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Number theory.

Explore the powers of divisibility, modular arithmetic, and infinity.

Yash Singhal
Munem Shahriar
Harsh Shrivastava
shivamani patil
Mark Buchanan
Mohammad Farhat
Matin Naseri
Shourya Pandey
Agnishom Chattopadhyay
Lee Care Gene
Anandmay Patel
Satyabrata Dash
Leon Macleod
Christopher Williams
Abdur Rehman Zahid
Akeel Howell
Daniel Schwartz
Chenjia Lin
Andrew Hayes

A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself. For example, 5 is a prime number because it has no positive divisors other than 1 and 5.

The first 49 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, and 227.

In contrast to prime numbers, a composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 4 is a composite number because it has three positive divisors: 1, 2, and 4. All positive integers greater than 1 are either prime or composite. 1 is the only positive integer that is neither prime nor composite.

Prime numbers are critical for the study of number theory . Nearly all theorems in number theory involve prime numbers or can be traced back to prime numbers in some way. Prime numbers are also important for the study of cryptography . The RSA method of encryption relies upon the factorization of a number into primes. Finally, prime numbers have applications in essentially all areas of mathematics. Prime numbers act as "building blocks" of numbers, and as such, it is important to understand prime numbers to understand how numbers are related to each other.

The Fundamental Theorem of Arithmetic

Identifying prime numbers, primality tests.

Distribution of Primes

Mersenne Primes

Number theory applications, additional examples.

Main Article: Fundamental Theorem of Arithmetic See Also: Prime Factorization

The fundamental theorem of arithmetic separates positive integers into two classifications: prime or composite.

Fundamental Theorem of Arithmetic Every integer greater than 1 is either prime (it has no divisors other than 1 and itself) or composite (it has more than two divisors). Furthermore, every integer greater than 1 has a unique prime factorization up to the order of the factors.

The prime factorization of a positive integer is that number expressed as a product of powers of prime numbers. Prime factorizations are often referred to as unique up to the order of the factors . This means that each positive integer has a prime factorization that no other positive integer has, and the order of factors in a prime factorization does not matter.

Give the prime factorization of 48. 48 is divisible by the prime numbers 2 and 3. The highest power of 2 that 48 is divisible by is $16=2^4.$ The highest power of 3 that 48 is divisible by is $3=3^1.$ Thus, the prime factorization of 48 is \[48=2^4 \times 3^1.\] The fundamental theorem of arithmetic guarantees that no other positive integer has this prime factorization. $_\square$

Prime factorization is the primary motivation for studying prime numbers. Many theorems, such as Euler's theorem , require the prime factorization of a number. Prime factorization can help with the computation of GCD and LCM . Prime factorization is also the basis for encryption algorithms such as RSA encryption . In order to develop a prime factorization, one must be able to efficiently and accurately identify prime numbers.

The simplest way to identify prime numbers is to use the process of elimination. List out numbers, eliminate the numbers that have a prime divisor that is not the number itself, and the remaining numbers will be prime. This process can be visualized with the sieve of Eratosthenes .

Sieve of Eratosthenes

Another way to Identify prime numbers is as follows:

First, choose a number, for example, 119.
Now, note that prime numbers between 1 and 10 are 2, 3, 5, 7. (Why between 1 and 10? This question is answered in the theorem below.) Divide the chosen number 119 by each of these four numbers.
If it's divisible by any of the four numbers, then it isn't a prime number; if it's not divisible by any of the four numbers, then it is prime.
119 is divisible by 7, so it is not a prime number.

What is the next term in the following sequence?

\[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots \]

Fortunately, one does not need to test the divisibility of each smaller prime to conclude that a number is prime. The number of primes to test in order to sufficiently prove primality is relatively small.

If $n$ is a composite number, then it must be divisible by a prime $p$ such that $p \le \sqrt{n}.$

Proof by Contradiction : Suppose that $n$ is a composite number, and it is only divisible by prime numbers that are greater than $\sqrt{n}.$ Let two of its factors be $q$ and $r,$ with $q,r > \sqrt{n}.$ Then $n=kqr,$ where $k$ is a positive integer. However, if $q$ and $r$ are both greater than $\sqrt{n},$ then $qr>n.$ This cannot be true, because $n=kqr,$ and $k$ is a positive integer. Thus, $n$ must be divisible by a prime that is less than or equal to $\sqrt{n}.\ _\square$

One can apply divisibility rules to efficiently check some of the smaller prime numbers. Long division should be used to test larger prime numbers for divisibility. It is helpful to have a list of prime numbers handy in order to know which prime numbers should be tested.

Is 211 a prime number? If 211 is a prime number, then it must not be divisible by a prime that is less than or equal to $\sqrt{211}.$ $\sqrt{211}$ is between 14 and 15, so the largest prime number that is less than $\sqrt{211}$ is 13. It is therefore sufficient to test 2, 3, 5, 7, 11, and 13 for divisibility. 211 is not divisible by any of those numbers, so it must be prime. $_\square$

What is the sum of the two largest two-digit prime numbers? If a two-digit number is composite, then it must be divisible by a prime number that is less than or equal to $\sqrt{100}=10.$ Therefore, it is sufficient to test 2, 3, 5, and 7 for divisibility. Counting backward, 99 is divisible by 3; 98 is divisible by 2; 97 is not divisible by 2, 3, 5, or 7, implying it is the largest two-digit prime number; 96 is divisible by 2; 95 is divisible by 5; 94 is divisible by 2; 93 is divisible by 3; 92 is divisible by 2; 91 is divisible by 7; 90 is divisible by 2; 89 is not divisible by 2, 3, 5, or 7, implying it is the second largest two-digit prime number. The sum of the two largest two-digit prime numbers is $97+89=186.$ $_\square$

What is the largest 3-digit prime number? If a a three-digit number is composite, then it must be divisible by a prime number that is less than or equal to $\sqrt{1000}.$ $\sqrt{1000}$ is between 31 and 32, so it is sufficient to test all the prime numbers up to 31 for divisibility. Counting backward, we have the following: 999 is the largest 3-digit number, but as it is divisible by $3$, it is not prime. 998 is the second largest 3-digit number, but as it is divisible by $2$, it is not prime. 997 is not divisible by any prime number up to $31,$ so it must be prime. $_\square$

Which prime number follows 1997? 1998 is divisible by 2. If 1999 is composite, then it must be divisible by a prime number that is less than or equal to $\sqrt{1999}$. $\sqrt{1999}$ is between 44 and 45, so the possible prime numbers to test are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, and 43. 1999 is not divisible by any of those numbers, so it is prime. $_\square$

Sometimes, testing a number for primality does not involve exhaustively searching for prime factors, but instead making some clever observation about the number that leads to a factorization. The next couple of examples demonstrate this.

In the following sequence, how many prime numbers are present? \[121,12321,1234321,123454321,\ldots\] We have \[\begin{align} 121&= 11×11\\ 12321&= 111×111\\ 1234321&= 1111×1111\\ 123454321&= 11111×11111. \end{align}\] So, no numbers in the given sequence are prime numbers. $_\square$

How many numbers in the following sequence are prime numbers? \[101,10201,102030201,1020304030201, \ldots\] We have the following factorization: $101$ has no factors other than 1 and itself. So, it is a prime number. $10201 = 101×101$. $102030201 = 10101×10101$. $1020304030201 = 1010101×1010101$. So, there is only $1$ prime number in the given sequence. $_\square$

Is 12345 a prime number? We have $\frac{12345}{5}=2469.$ So 12345 is divisible by 5 and therefore is not prime. $_\square$

Main Article: Primality Testing

There are other methods that exist for testing the primality of a number without exhaustively testing prime divisors. These methods are called primality tests . One of these primality tests applies Wilson's theorem .

Wilson's Theorem A positive integer $p>1$ is prime if and only if \[(p-1)! \equiv -1 \pmod{p}.\]

Show that 7 is prime using Wilson's theorem. We have \[\begin{align} 6!&=720\\ 720 &\equiv -1 \pmod{7}. \end{align}\] Therefore, 7 must be prime. $_\square$

Testing primes with this theorem is very inefficient, perhaps even more so than testing prime divisors. However, this theorem does give insight that a number's primality is not linked purely to the divisors of that number. There are other "traces" in a number that can indicate whether the number is prime or not.

Most primality tests are probabilistic primality tests . These kinds of tests are designed to either confirm that the number is composite, or to use probability to designate a number as a probable prime . A probable prime is a number that has been tested sufficiently to give a very high probability that it is prime. An example of a probabilistic prime test is the Fermat primality test, which is based on Fermat's little theorem .

Fermat Primality Test Given a positive integer $n,$ Choose a positive integer $a>1$ at random that is coprime to $n$. Compute $a^{n-1} \bmod {n}.$ If the result is not $1,$ then $n$ is composite. Otherwise, $n$ may or may not be prime. If the result is $1$ at this step, but $n$ is composite, then $n$ is pseudoprime to base $a.$ Repeat these steps any number of times. Each repetition of these steps improves the probability that the number is prime. However, this process can never guarantee that a number is prime. A composite number that gets a result of $1$ in the second step for all $a$ less than $n$ is called a Carmichael number .

Show that 91 is composite using the Fermat primality test with the base $a=2$. The goal is to compute $2^{90}\bmod{91}.$ Use the method of repeated squares . Compute 90 in binary: \[90_{10}=1011010_{2}.\] Compute the residues of the repeated squares of 2: \[\begin{align} 2^{2^0} &\equiv 2 \pmod{91} \\ 2^{2^1} &\equiv 4 \pmod{91} \\ 2^{2^2} &\equiv 16 \pmod{91} \\ 2^{2^3} &\equiv 74 \pmod{91} \\ 2^{2^4} &\equiv 16 \pmod{91} \\ 2^{2^5} &\equiv 74 \pmod{91} \\ 2^{2^6} &\equiv 16 \pmod{91} \\ &\vdots\\ 2^{90} &= 2^{2^6} \times 2^{2^4} \times 2^{2^3} \times 2^{2^1} \\\\ 2^{90} &\equiv (16)(16)(74)(4) \pmod{91} \\ &\equiv 64 \pmod{91}. \end{align}\] The result is not $1.$ Therefore, $91$ is not prime. $_\square$

This process might seem tedious to do by hand, but a computer could perform these calculations relatively efficiently. Thus, the Fermat primality test is a good method to screen a large list of numbers and eliminate numbers that are composite. Then, a more sophisticated algorithm can be used to screen the prime candidates further.

In general, identifying prime numbers is a very difficult problem. This, along with integer factorization, has no algorithm in polynomial time. In fact, it is so challenging that much of computer cryptography is built around the fact that there is no known computationally feasible way to find the factors of a large number.

Main Article: Distribution of Primes See Also: Infinitely Many Primes

It has been known for a long time that there are infinitely many primes . However, the question of how prime numbers are distributed across the integers is only partially understood. The prime number theorem gives an estimation of the number of primes up to a certain integer.

Prime Number Theorem Let $\pi(x)$ be the prime counting function . For any real number $x,$ $\pi(x)$ gives the number of prime numbers that are less than or equal to $x.$ Then \[\lim_{x \rightarrow \infty} \frac{\hspace{2mm} \pi(x)\hspace{2mm} }{\frac{x}{\ln{x}}}=1.\] This implies that for sufficiently large $x,$ \[\pi(x) \sim \frac{x}{\ln{x}}.\]

There are many open questions about prime gaps. A prime gap is the difference between two consecutive primes. For example, the prime gap between 13 and 17 is 4. Bertrand's postulate gives a maximum prime gap for any given prime.

Bertrand's Postulate For any integer $n>3,$ there always exists at least one prime number $p$ such that \[n<p<2n-2.\] This implies that for the $k^\text{th}$ prime number, $p_k,$ the next consecutive prime number is subject to \[p_{k+1}<2p_k.\]

This is, unfortunately, a very weak bound for the maximal prime gap between primes. Prime gaps tend to be much smaller, proportional to the primes. For example, the first occurrence of a prime gap of at least 100 occurs after the prime 370261 (the next prime is 370373, a prime gap of 112).

The most famous problem regarding prime gaps is the twin prime conjecture . This conjecture states that there are infinitely many pairs of primes for which the prime gap is 2, but as of this writing, no proof has been discovered.

Another famous open problem related to the distribution of primes is the Goldbach conjecture . This conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.

One of the most significant open problems related to the distribution of prime numbers is the Riemann hypothesis . Although the Riemann hypothesis has wide-reaching implications in number theory, Riemann's original motivation for formulating the conjecture was to better understand the distribution of prime numbers. The Riemann hypothesis relates the real parts of the zeros of the Riemann zeta function to the oscillations of the prime numbers about their "expected" positions given the estimation of the prime counting function above.

Main Article: Mersenne Primes

A Mersenne prime is a prime that can be expressed as $2^p-1,$ where $p$ is a prime number.

The first five Mersenne primes are listed below:

\[\begin{array}{c|rr} p & 2^p-1= & M_p\\ \hline 2 & 2^2-1= & 3 \\ 3 & 2^3-1= & 7 \\ 5 & 2^5-1= & 31 \\ 7 & 2^7-1= & 127 \\ 13 & 2^{13}-1= & 8191 \end{array}\]

Note that having the form of $2^p-1$ does not guarantee that the number is prime. $2^{11}-1=2047$ is not a prime number; its prime factorization is $23 \times 89.$

From the list above, it might seem as though Mersenne primes are relatively easy to find by simply plugging in prime numbers into $2^p-1$. However, Mersenne primes are exceedingly rare. As of January 2018, only 50 Mersenne primes are known, the largest of which is $2^{77,232,917}-1$. This number is also the largest known prime number. In fact, many of the largest known prime numbers are Mersenne primes. This is due to the Lucas-Lehmer primality test , which is an efficient algorithm that is specific to testing primes of the form $2^p-1$. Although Mersenne primes continue to be discovered, it is an open problem whether or not there are an infinite number of them.

Another notable property of Mersenne primes is that they are related to the set of perfect numbers . A perfect number is a positive integer that is equal to the sum of its proper positive divisors. Each Mersenne prime corresponds to an even perfect number:

Let $M_p$ be a Mersenne prime. Then $\frac{M_p\big(M_p+1\big)}{2}$ is an even perfect number. Furthermore, all even perfect numbers have this form.

Give the perfect number that corresponds to the Mersenne prime 31. The perfect number is given by the formula above: \[\frac{(31)(32)}{2}=496.\] This number can be shown to be a perfect number by finding its prime factorization: \[496=2^4\times 31.\] Then listing out its proper divisors gives \[\text{proper divisors of 496}=\{1,2,4,8,16,31,62,124,248\}.\] Summing these divisors, we have \[1+2+4+8+16+31+62+124+248=496.\ _\square\]

One of the most fundamental theorems about prime numbers is Euclid's lemma.

Euclid's Lemma Let $p$ be prime. If $p \mid ab$, then $p \mid a$ or $p \mid b$.

By Contradiction: Suppose $p$ does not divide $a$. Since the only divisors of $p$ are $1$ and $p,$ and $p$ doesn't divide $a,$ we must have $\gcd (a, p) =1.$ By Bezout's identity , there exist some $u$ and $v$ such that $ua+vp=1$. Multiplying both sides of this equation by $b$ gives $b=uab+vpb$. Now $p$ divides $uab$ $($since it is given that $p \mid ab),$ and $p$ also divides $vpb$. Therefore, $p$ divides their sum, which is $b$. Without loss of generality, if $p$ does not divide $b,$ then it must divide $a.$ $ _\square $

Euclid's lemma can seem innocuous, but it is incredibly important for many proofs in number theory. For example, it is used in the proof that the square root of 2 is irrational .

Prime factorizations can be used to compute GCD and LCM .

GCD and LCM using Prime Factorizations Given positive integers $m$ and $n,$ let their prime factorizations be given by \[\begin{align} m&=p_1^{j_1} \times p_2^{j_2} \times p_3^{j_3} \times \cdots\\ n&=p_1^{k_1} \times p_2^{k_2} \times p_3^{k_3} \times \cdots, \end{align}\] where $p_1, p_2, p_3, \ldots$ are distinct primes and each $j_i$ and $k_i$ are integers. Then the GCD of these integers is given by \[\gcd(m,n)=p_1^{\min(j_1,k_1)} \times p_2^{\min(j_2,k_2)} \times p_3^{\min(j_3,k_3)} \times \cdots,\] and the LCM of these integers is given by \[\text{lcm}(m,n)=p_1^{\max(j_1,k_1)} \times p_2^{\max(j_2,k_2)} \times p_3^{\max(j_3,k_3)} \times \cdots.\]

Using prime factorizations, what are the GCD and LCM of 36 and 48? The prime factorizations are \[\begin{align} 36 &= 2^2 \times 3^2 \\ 48 &= 2^4 \times 3^1. \end{align}\] Each number has the same primes, 2 and 3, in its prime factorization. The GCD is given by taking the minimum power for each prime number: \[\begin{align} \gcd(36,48) &= 2^{\min(2,4)} \times 3^{\min(2,1)} \\ &= 2^2 \times 3^1 \\ &= 12. \end{align}\] The LCM is given by taking the maximum power for each prime number: \[\begin{align} \text{lcm}(36,48) &= 2^{\max(2,4)} \times 3^{\max(2,1)} \\ &= 2^4 \times 3^2 \\ &= 144.\ _\square \end{align}\]

Prime numbers are important for Euler's totient function .

Given a positive integer $n$, Euler's totient function , denoted by $\phi(n),$ gives the number of positive integers less than $n$ that are co-prime to $n.$

What is $\phi(10)?$ Listing out the positive integers that are less than 10 gives \[\{1,2,3,4,5,6,7,8,9\}.\] Of those numbers, list the subset of numbers that are co-prime to 10: \[\{1,3,7,9\}.\] This set contains 4 elements. Therefore, $\phi(10)=4.\ _\square$

If $n$ is a power of a prime, then Euler's totient function can be computed efficiently using the following theorem:

For any given prime $p$ and positive integer $n$, \[\phi(p^n)=p^n-p^{n-1}.\]

Then, the value of the function for products of coprime integers can be computed with the following theorem:

Given co-prime positive integers $m$ and $n$, \[\phi(mn)=\phi(m) \times \phi(n).\]

The consequence of these two theorems is that the value of Euler's totient function can be computed efficiently for any positive integer, given that integer's prime factorization.

What is $\phi(48)?$ The prime factorization of 48 is \[48=2^4 \times 3^1.\] Then, \[\begin{align} \phi(2^4) &= 2^4-2^3=8 \\ \phi(3^1) &= 3^1-3^0=2 \\ &\vdots\\ \phi(48) &= 8 \times 2=16.\ _\square \end{align}\]

Euler's totient function is critical for Euler's theorem .

Euler's Theorem Let $a$ and $n$ be coprime integers with $n>0$. Then, \[a^{\phi(n)} \equiv 1 \pmod{n}.\]

If $n$ is a prime number, then this gives Fermat's little theorem .

Fermat's Little Theorem Let $p$ be a prime number and let $a$ be an integer coprime to $p.$ Then, \[a^{p-1} \equiv 1 \pmod{p}.\]

The properties of prime numbers can show up in miscellaneous proofs in number theory .

Let $p$ be a prime number greater than $3$. Prove that $p^2-1$ is always divisible by 6. Any integer can be written in the form $6k+n,\ n \in \{0,1,2,3,4,5\}$. $6k$ cannot be a prime. $6k+1$ may be a prime. $6k+2$ cannot be a prime. $6k+3$ cannot be a prime. $6k+4$ cannot be a prime. $6k+5$ may be a prime. Thus, any prime $p > 3$ can be represented in the form $6k+5$ or $6k+1$. $p^2-1$ can be factored to $(p+1)(p-1).$ Case 1 : $p=6k+1$ If this is the case, $p^2-1=(6k+2)(6k),$ which implies $6 \mid (p^2-1).$ Case 2 : $p=6k+5$ If this is the case, $p^2-1=(6k+6)(6k+4),$ which implies $6 \mid (p^2-1).$ One of the factors, $p-1$ or $p+1$, will be divisible by $6$. Thus, $p^2-1$ is always divisible by $6$. $_\square$

$n$ is the greatest prime number less than 50, and $m$ is the least prime number greater than 50. What is the value of $n+m?$ Let's work backward for $n$. $49$ is divisible by $7$, and from the property of primes it is enough information to conclude that the number is not prime. $48$ is divisible by $2,$ so cancel it. When we look at $47,$ it doesn't have any divisor other than one and itself. So it is indeed a prime: $n=47.$ We use the same process in looking for $m$. $51$ is divisible by $3$. $52$ is divisible by $2$. $53$ doesn't have any other divisor other than one and itself, so it is indeed a prime: $m=53.$ Therefore, \[n+m=100.\ _\square\]

Consider the expression $2^{n}-1,$ where $n$ is an integer greater than $1.$ What are the least two values of $n$ for which the expression doesn't produce a prime number? Let's check by plugging in numbers in increasing order. $2^{2}-1=3$ is prime. $2^{3}-1=7$ is prime. $2^{4}-1=15$, which is divisible by 3, so it isn't prime. $2^{5}-1=31$ is prime. $2^{6}-1=63$, which is divisible by 7, so it isn't prime. Therefore, the least two values of $n$ are 4 and 6. $_\square$

An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversible prime may be used to mean the same as emirp, but may also, ambiguously, include the palindromic primes.

The sequence of emirps begins 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, ... (sequence A006567 in the OEIS).

All non-palindromic permutable primes are emirps.

As of November 2009, the largest known emirp is 1010006+941992101×104999+1, found by Jens Kruse Andersen in October 2007.

The term 'emirpimes' (singular) is used also in places to treat semiprimes in a similar way. That is, an emirpimes is a semiprime that is also a (distinct) semiprime upon reversing its digits.

Prime Factorization
Divisibility Rules
Fundamental Theorem of Arithmetic
Chinese Remainder Theorem
Mersenne Prime
RSA Encryption

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Prime Numbers Chart and Calculator

A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

(if we can make it by multiplying other whole numbers it is a Composite Number )

Here we see it in action:

Here is a list of all the prime numbers up to 1,000:

Is 8 a Prime Number? No , because it can be made by 2×4=8
Is 73 a Prime Number? Yes , as no other whole numbers multiply together to make it.

Calculator ... Is It Prime?

Find out if a number is Prime or not (works on numbers up to 4,294,967,295):

$SplashLearn$

Prime Numbers – Definition, Chart, Examples, Practice Problems

What are prime numbers, difference between prime number and composite number, how to find prime numbers, solved examples on prime numbers, practice problems on prime numbers.

Frequently Asked Questions on Prime Numbers

Prime numbers are numbers greater than 1 that only have two factors, 1 and the number itself. This means that a prime number is only divisible by 1 and itself. If you divide a prime number by a number other than 1 and itself, you will get a non-zero remainder.

Numbers that have more than 2 factors (but finite number of factors) are known as composite numbers .

10 and 100 More than the Same Number Game

Prime Numbers Definition

A prime number can be defined as a natural number greater than 1 whose only factors are 1 and the number itself.

A prime number is a positive integer greater than 1 that cannot be written as a product of two distinct integers which are greater than 1.

Related Worksheets

$10 and 100 More than a 3-digit Number$

Examples and Non-examples of Prime Numbers

Properties of prime numbers.

Prime numbers are natural numbers greater than 1.
2 is the smallest prime number.
2 is the only even prime number. All the prime numbers except 2 are odd.
Any two prime numbers are always coprime.
Any composite number can be uniquely expressed as the product of its prime factors.
A prime number has only two factors $— 1$ and the number itself.
If two prime numbers have only 1 composite number between them, they are called twin-prime numbers.
Every even positive integer greater than 2 can be written as the sum of two prime numbers.
Every positive integer greater than 1 has at least one prime factor.

The Sieve of Eratosthenes

In the third century B.C. the Greek mathematician, Eratosthenes, found a very simple method of finding the prime numbers.

Follow the given steps to identify the prime numbers between 1 and 100.

Step 1: Make a hundreds chart. (Write all the natural numbers between 1 to 100 using 10 rows and 10 columns.)

Step 2: Leave 1 as it is neither a prime number nor composite number.

Step 3: Encircle 2 and cross out all its multiples (such as 4, 6, 8, 10, and so on) as they are not prime.

Step 4: Encircle the next uncrossed number which is 3 and cross out all its multiples . Ignore the previously crossed out numbers like 6, 12 and 18 and so on.

Step 5: Continue the process of encircling the next uncrossed number and crossing out its multiples till all the numbers in the table are either encircled or crossed except 1.

$The Sieve of Eratosthenes$

So, from the table it is clear that 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers.

There are 25 prime numbers between 1 and 100.

1) Every prime number (except 2 and 3) can be written in the form of 6n + 1 or 6n – 1.

To check whether a given number is prime or not, you can simply check if it can be written in the form $6n + 1$ or $6n \;-\; 1$.

Examples: 13 can be expressed as $6(2) + 1$. Thus, 13 is a prime number.

$6(1) \;–\; 1 = 5$

$6(1) + 1 = 7$

$6(2)\; –\; 1 = 11$

$6(2) + 1 = 13$

$6(3) \;–\; 1 = 17$

$6(3) + 1 = 19$

(Note that we do not consider the multiples of prime numbers when we use this method. For example, 25 and 49 can be written as $25 = 6(4) + 1,\; 49 = 6(8) \;-\; 1$, but they are not prime numbers.)

2) To find the prime numbers greater than 40, we can use this method.

If we can express a number greater than 40 in the form $n^{2} + n + 41$, where $n = 0,\; 1,\; 2,\; …..,\; 39$, then it is a prime number.

$(0)^{2} + 0 + 0 = 41$

$(1)^{2} + 1 + 41 = 43$

$(3)^{2} + 3 + 41 = 53$

3) You can use the factorization method and find the number of factors the number has. If it has 2 unique factors – 1 and itself, it is a prime number. Otherwise, it is composite.

For example, to find the factors of n, divide n by each natural number up to n. If the remainder is 0, it is a factor.

Co-Prime Numbers and Twin Prime Numbers

Co-Primes: Two numbers are said to be co-primes if they have only 1 common factor, that is, 1. It is not necessary for these numbers to be prime numbers. For example, 9 and 10 are co-primes . Let’s verify.

$finding common factors of 9 and 10 using factor tree$

Note that any two prime numbers are always co-prime. This is because out of their two factors, the common factor can only be 1. So, (3, 5), (11, 19) are some examples of co-primes.

Twin-Primes: A pair of prime numbers are known as twin primes if there is only one composite number between them. For example, (3, 5), (5, 7), (11, 13), (17, 19), etc.

Prime Numbers List

Take a look at the ‘prime numbers charts’ and check out the organized lists of prime numbers in the given range.

List of Prime Numbers between 1 and 100

Prime numbers between 1 to 100 are as follows:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

List of Prime Numbers between 1 and 200

There are 46 prime numbers between 1 and 200.

List of Prime Numbers between 1 and 1000

Facts about Prime Numbers

2 is the only prime number which is even.
2 and 3 are the only consecutive prime numbers.
Except for 0 and 1, a whole number is either a prime number or a composite number.
No prime number greater than 5 ends in a 5.
Sieve of Eratosthenes is one of the earliest methods of finding prime numbers.
Prime numbers get more rare as the number gets bigger.
There is no largest prime number. The largest known prime number (as of September 2021) is $2^{82,589,933} \;−\; 1$, a number which has 24,862,048 digits when written in base 10. By the time you read this it may be even larger.

In this article, we learned about prime numbers, their properties, methods to find prime numbers, and different lists of prime numbers. Let’s solve a few examples and practice problems based on these concepts for better understanding.

Example 1: Classify the given numbers as prime numbers or composite numbers.

13, 48, 49, 23, 74, 80, 71, 59, 45, 47

Example 2: Express 21 as the sum of two prime numbers.

Solution:

We can write 21 as

$21 = 19 + 2$

Here, 2 and 19 are both prime numbers.

Example 3: What prime numbers are there between 20 and 30?

The prime numbers between 20 and 30 are 23 and 29.

Example 4: What is the greatest prime number between 80 and 90?

The prime numbers between 80 and 90 are 83 and 89.

So, 89 is the greatest prime number between 80 and 90.

Prime Numbers

Attend this Quiz & Test your knowledge.

Which of the following is not a prime number?

What is the 10 th prime number, how many prime numbers are there between 40 and 50, which is the smallest odd prime number, which of the following pairs of numbers are co-prime, f requently asked questions on prime numbers.

Is 1 a prime number?

No, 1 is neither a prime number nor a composite number.

Can a prime number be negative?

No, prime numbers cannot be negative. Prime numbers are natural numbers greater than 1.

Why is 2 the only even prime number?

All even numbers larger than 2 are multiple of 2. So, 2 is the only even prime number.

What is the difference between prime numbers and co-prime numbers?

A prime number is a natural number greater than 1 that has only two factors – 1 and the number itself. A pair of numbers whose HCF is 1 (the only common factor is 1) are called co-prime numbers.

Which is the largest known prime number?

The largest known prime number (as of September 2021) is 2 82,589,933 − 1, a number which has 24,862,048 digits. By the time you read this it may be even larger than this.

What is the lowest prime number?

The smallest prime number is 2.

What makes a number prime?

If a number has only two factors (1 and itself), it is a prime number. If it has more than two factors, it is composite.

How to check if a number is prime?

Find the factors of the given number. Check the number of factors. If there are more than 2 factors, the number is not prime. If there are only two factors (1 and number itself), the number is prime.

What are the first ten prime numbers?

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Why 0 and 1 are neither prime nor composite?

Any number multiplied by 0 results in 0. So, 0 has infinitely many factors. However, a composite number can have only a finite number of factors. Also, $0 \lt 1$ and prime numbers are natural numbers greater than 1.

1 has only 1 factor, itself. So, 0 and 1 do not fit into the definitions of prime & composite numbers.

What is the total number of prime numbers up to 100?

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Prime Number Calculator

What is a prime number, how to use our prime number calculator, how to check if a number is prime or composite, how to find prime numbers, why do prime numbers matter, relatively prime numbers.

With our prime number calculator you can check if any given number is prime or composite. Keep reading if you want to learn what a prime number is, and how to check if a number is prime. We also discuss how to find prime numbers, as well as what relatively prime numbers are. As a bonus, we tell you if 1 is prime, and also what the oddest prime number is.

A natural number greater than 1 is called prime if it has exactly two factors, i.e., if the number is divisible only by 1 and itself.

When a natural number is greater than 1 and isn't prime, then it's called a composite number.

What about 1? 1 is neither prime nor composite as it has only one factor (itself).

Let's discuss some examples:

7 is prime because its only factors are 1 and 7. Indeed, none of the number preceding 7 (2, 3, 4, 5, and 6) is a factor of it, because none of these numbers divide 7 without a remainder. If you don't quite remember what that means, go to the remainder calculator .
8 is composite because 2 is a factor of 8, and so 8 has more factors than just 1 and 8.

Fun fact. The oddest number amongst primes is 2, as it is the only even prime: all other primes are odd!

Well, this prime number calculator is not a very complicated one. Just enter the number you wish to check into the calculator and voila! your answer will be shown below.

If your number is composite, the calculator will tell you its smallest non-trivial factor (non-trivial here means a factor greater than one).

The easiest way to verify that a given integer n is prime is to apply the so-called trial division algorithm: it consists of testing whether n is divisible by any number between 2 and n-1 . That's a lot of computations. Fortunately, the number of trials can be reduced; it is sufficient to check only the divisibility of n by prime numbers which do not exceed √n . A version of the trial division algorithm powers this prime number calculator.

Well, you cannot find every prime number, because Euclid proved sometime around 300 BC that there are an infinite number of them. If you want to find all prime numbers up to some given limit, n , you may resort to the algorithm known as the Sieve of Eratosthenes :

Write down all numbers from 2 to n
Start with the smallest number in our list: 2 . Circle 2 and cross out all consecutive multiples of 2 (i.e., 4, 6, 8, ..., 120 )
Take the smallest number that is not circled or crossed out: 3 . Circle it and cross out all its further multiples: 3, 6, 9, ...
Continue in this same way: find the smallest available number p , circle it and cross out all of its consecutive multiples
Check if there are numbers greater than p not yet crossed out. If so, repeat Step 4. If no, we're done.

The circled numbers in the list are all the primes below n .

The main idea here is that every number assigned to p in any step of the algorithm is necessarily prime; otherwise, it would be crossed out as a multiple of some smaller, already circled, prime number.

In the animation below you see the Sieve of Eratosthenes searching for all primes up to 120 . Here's the color legend:

red: multiples of 2
green: multiples of 3
blue: multiples of 5
yellow: multiples of 7
the remaining purple numbers are prime

As you may imagine, for REALLY BIG numbers it is quite challenging to decide whether they are prime or composite, and therefore very elaborate methods are required. The search for big primes is on-going: as of March 2020, the largest known prime number has 24,862,048 digits! Our time unit converter helps us estimate that it would take roughly 288 days (so more than three-quarters of a year) to write this number down in its entirety, provided that one digit per second is being written down!

Prime numbers are crucial in number theory because of the fundamental theorem of arithmetic : every natural number greater than 1 can be written as a product of primes in a unique way (up to the order of multiplication). In other words, primes can be thought of as the building blocks of all other natural numbers. To learn more about prime factorization, check our prime factorization calculator .

Actually, this theorem is one of the main reasons why we don't want 1 to be called a prime number. If 1 were prime, then prime factorization would no longer be unique, because we would have, e.g., 6 = 2 * 3 = 1 * 2 * 3 = 1 * 1 * 2 * 3 = ...

As for real-life applications of prime numbers, you may encounter them, e.g., in cryptography protocols, such as the well-known RSA encryption .

Do not confuse the notion of prime numbers with that of relatively prime numbers! Two natural numbers are called relatively prime (or coprime ) if there is no integer other than 1 that divides both these numbers. In other words, their Greatest Common Factor (GCF) is equal to 1.

For instance:

18 and 30 are not relatively prime, as they are both divisible by 3 . You may also note that their GCD is 6 .
18 and 35 are relatively prime: the factors of 18 read 1, 2, 3, 6, 9, 18 and that of 35 read 1, 5, 7, 35 . We see that the sole common divisor of 18 and 35 is 1 .

To learn more, visit the GCF calculator .

Two primes are always relatively prime
But numbers need not be prime in order to be relatively prime!

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What Is a Prime Number? How to Tell If a Number Is Prime

A prime number is a natural number that can only be divided, without a remainder, by itself and 1. In other words, a prime number has exactly two factors. For example, 13 is only divisible by 13 and 1. In contrast, a composite number is a natural number that can be divided evenly by any number besides itself and 1. A composite number has more than two factors. For example, 14 is divisible by 1, 2, 7, and 14.

Here is a list of the prime numbers up to 1000 and a look at how to tell if a number is prime.

Interesting Prime Number Facts

The state of being prime is called primality .
There are an infinite number of prime numbers.
Zero and one are not prime numbers.
Two is the only even prime number.
Two and three are the only consecutive prime numbers.
No prime number greater than five ends in 5.
No prime number ends with 0.
Goldbach Conjecture : Every even integer greater than 2 can be expressed as the sum of two prime numbers.
Every prime number greater than 2 and 3 can be represented as 6n+1 or 6n-1.
Prime Number Theorem : The probability that a number is prime is inversely proportional to its number of digits.
Lemoine’s Conjecture : Any odd integer greater than 5 can be expressed as the sum of an off prime and an even semiprime. A semiprime is the product of two prime numbers.

Prime Numbers Up to 1000

The smallest prime number is 2, which is also the only even prime number. Here is a table of all the prime numbers up to 1000.

Is 1 a Prime Number?

The number 1 is not usually considered a prime number. It’s also not a composite number.

1 is not a prime number because it does not have exactly two positive factors.
1 is not a composite number because it does not have more than two factors.

Note: There are some people who argue 1 is a prime number because it’s divisible by itself and 1 (even though these two values are the same thing).

How to Tell If a Number Is Prime

There are a few different ways to tell whether or not a number is prime. The methods are called primality tests , even though some of them actually test whether a number is composite.

Basically, you test whether a number n is evenly divisible by any prime number between 2 and √ n . This is called trial division or factorization.

No even number except 2 is prime. If a number ends with 0, 2, 4, 6, or 8, it’s a composite number.
If the sum of the digits of a number are divisible by 3, it’s a composite number. A prime number can end with 3.
No prime number ends with 5, except 5.
If a number passes all of these tests, check to see if it’s divisible by prime numbers smaller than it. It’s not necessary to check prime numbers greater than √ n . Start with 3, 5, 7, 11, and work your way up to √ n .
Check whether or not a number can be expressed as either 6n+1 or 6n-1. For example, the prime number 11 can be written as 6(2)-1.

Examples: Finding a Prime Number Using Factorization

Is 15874 prime?
Right away, you can see it’s not prime because it ends with an even number.
Is 26577 a prime number?
It does not end in 0, 2, 4, 6, 8.
The sum of the digits 2 + 6 + 5 + 7 + 7 = 27.
27 is divisible by 3, so 26577 is not prime.
Is 103 a prime number?
It does not end in 5.
The sum of the digits 1 + 0 + 3 = 4. It is not divisible by 3.
The √ 103 is ~10.14. So, check to see if 103 is divisible by other primes under 10.
103 is not evenly divisible by 7.
103 is a prime number!

What Is the Largest Prime Number?

There are an infinite number of prime numbers, so computers discover new primes (slowly, because it takes a lot of computing power). To date, the largest prime number is 2 82,589,933 -1. The Great Internet Mersenne Prime Search (GIMPS) found this prime on December 7, 2018.

Adler, Irving (1960). The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space . Golden Press.
Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (2nd ed.). Springer. ISBN 0-387-25282-7.
Dudley, Underwood (1978). “ Section 2: Unique factorization “. Elementary Number Theory (2nd ed.). W.H. Freeman and Co. ISBN 978-0-7167-0076-0.
“ GIMPS Project Discovers Largest Known Prime Number: 2 82,589,933 -1 “. Mersenne Research, Inc .
Ziegler, Günter M. (2004). “The great prime number record races”. Notices of the American Mathematical Society . 51 (4): 414–416.

Prime Nu...

Prime Numbers

What is a prime number.

A prime number is any natural number (counting number) that is greater than 1 and is divisible only by 1 and itself. Examples of prime numbers - 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , 7 \hspace{0.2em} 7 \hspace{0.2em} 7 , 11 \hspace{0.2em} 11 \hspace{0.2em} 11 , 13 \hspace{0.2em} 13 \hspace{0.2em} 13 , 17 \hspace{0.2em} 17 \hspace{0.2em} 17 , etc.

So, a prime number has only two positive factors – 1 \hspace{0.2em} 1 \hspace{0.2em} 1 and itself. And this is the property that makes prime numbers so special. More on that later.

What Is a Composite Number?

Counting numbers that have more than two positive factors are known as composite numbers. As such, they can be expressed as a product of smaller factors. For example, 4 \hspace{0.2em} 4 \hspace{0.2em} 4 , 10 \hspace{0.2em} 10 \hspace{0.2em} 10 , 15 \hspace{0.2em} 15 \hspace{0.2em} 15 , etc.

So basically every natural number greater than 1 \hspace{0.2em} 1 \hspace{0.2em} 1 is either prime or composite - but never both. 1 \hspace{0.2em} 1 \hspace{0.2em} 1 is neither prime nor composite.

What Are the Prime Numbers From 1 to 100?

If you want to be good at math, you have to be good with numbers. And being able to quickly identify whether a number is prime or not is one of the things that help you be great with numbers.

There are a total of 25 \hspace{0.2em} 25 \hspace{0.2em} 25 prime numbers between 1 \hspace{0.2em} 1 \hspace{0.2em} 1 and 100 \hspace{0.2em} 100 \hspace{0.2em} 100 . Make sure you familiarize yourself with each of them.

So the prime numbers up to 100 \hspace{0.2em} 100 \hspace{0.2em} 100 are -

2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , 7 \hspace{0.2em} 7 \hspace{0.2em} 7 , 11 \hspace{0.2em} 11 \hspace{0.2em} 11 , 13 \hspace{0.2em} 13 \hspace{0.2em} 13 , 17 \hspace{0.2em} 17 \hspace{0.2em} 17 , 19 \hspace{0.2em} 19 \hspace{0.2em} 19 , 23 \hspace{0.2em} 23 \hspace{0.2em} 23 , 29 \hspace{0.2em} 29 \hspace{0.2em} 29 , 31 \hspace{0.2em} 31 \hspace{0.2em} 31 , 37 \hspace{0.2em} 37 \hspace{0.2em} 37 , 41 \hspace{0.2em} 41 \hspace{0.2em} 41 , 43 \hspace{0.2em} 43 \hspace{0.2em} 43 , 47 \hspace{0.2em} 47 \hspace{0.2em} 47 , 53 \hspace{0.2em} 53 \hspace{0.2em} 53 , 59 \hspace{0.2em} 59 \hspace{0.2em} 59 , 61 \hspace{0.2em} 61 \hspace{0.2em} 61 , 67 \hspace{0.2em} 67 \hspace{0.2em} 67 , 71 \hspace{0.2em} 71 \hspace{0.2em} 71 , 73 \hspace{0.2em} 73 \hspace{0.2em} 73 , 79 \hspace{0.2em} 79 \hspace{0.2em} 79 , 83 \hspace{0.2em} 83 \hspace{0.2em} 83 , 89 \hspace{0.2em} 89 \hspace{0.2em} 89 , and 97 \hspace{0.2em} 97 \hspace{0.2em} 97 .

How to Tell If a Number Is Prime?

Now, of course, you can't memorize all of the prime numbers. There are infinitely many of them. So wouldn't it be great if there was a simple way of checking whether a number is prime or not? Well, there is!

Let me explain with an example.

Is 97 \hspace{0.2em} 97 \hspace{0.2em} 97 a prime number?

Here's how you find out if a number is prime.

Step 1. Make sure it is a counting number greater than 1 \hspace{0.2em} 1 \hspace{0.2em} 1 . And that it isn't a perfect square. Or else, the number cannot be prime.

97 \hspace{0.2em} 97 \hspace{0.2em} 97 passes this test.

Step 2. Think of the largest number whose square is less than the given number.

is 100 \hspace{0.2em} 100 \hspace{0.2em} 100 (more than 97 \hspace{0.2em} 97 \hspace{0.2em} 97 ).

Step 3. List the prime numbers up to the number found in the previous step.

Prime numbers up to 9 \hspace{0.2em} 9 \hspace{0.2em} 9 are - 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , and 7 \hspace{0.2em} 7 \hspace{0.2em} 7 .

Step 4. Is the given number divisible by any of the prime numbers from the step above? If yes, the number isn't prime. If not, it is prime.

97 isn't divisible by 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , or 7 \hspace{0.2em} 7 \hspace{0.2em} 7 . Hence, 97 \hspace{0.2em} 97 \hspace{0.2em} 97 is prime.

I know. It appears like too many steps with a lot to do. But give yourself some time and practice. And you'll see it's much simpler than it looks. More so, if you know the common divisibility rules and the square of numbers up to 20 \hspace{0.2em} 20 \hspace{0.2em} 20 .

Any number with two or more digits whose rightmost digit is 0 \hspace{0.2em} 0 \hspace{0.2em} 0 , 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 4 \hspace{0.2em} 4 \hspace{0.2em} 4 , 6 \hspace{0.2em} 6 \hspace{0.2em} 6 , 8 \hspace{0.2em} 8 \hspace{0.2em} 8 , or 5 \hspace{0.2em} 5 \hspace{0.2em} 5 cannot be prime. Reason? It will certainly be divisible by 2 \hspace{0.2em} 2 \hspace{0.2em} 2 or 5 \hspace{0.2em} 5 \hspace{0.2em} 5 . For example - 74 \hspace{0.2em} 74 \hspace{0.2em} 74 , 130 \hspace{0.2em} 130 \hspace{0.2em} 130 , 375 \hspace{0.2em} 375 \hspace{0.2em} 375 , etc.

Is 141 \hspace{0.2em} 141 \hspace{0.2em} 141 a prime number?

Step 1. 141 \hspace{0.2em} 141 \hspace{0.2em} 141 is a counting number greater than 1 \hspace{0.2em} 1 \hspace{0.2em} 1 and it isn't a perfect square. So we can move to the next step.

Step 2. The largest number whose square is less than 141 \hspace{0.2em} 141 \hspace{0.2em} 141 is 11 \hspace{0.2em} 11 \hspace{0.2em} 11 .

Step 3. The prime numbers up to 11 \hspace{0.2em} 11 \hspace{0.2em} 11 are 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , 7 \hspace{0.2em} 7 \hspace{0.2em} 7 , and 11 \hspace{0.2em} 11 \hspace{0.2em} 11 .

Step 4. Is 141 \hspace{0.2em} 141 \hspace{0.2em} 141 a prime number. That's it.

So, 141 a prime number. That's it.

3 \hspace{0.2em} 3 \hspace{0.2em} 3 has a simple divisibility test. If the sum of digits of a number is divisible by 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , the number must also be divisible by 3 \hspace{0.2em} 3 \hspace{0.2em} 3 . Try with 141 \hspace{0.2em} 141 \hspace{0.2em} 141 .

What Are Co-Prime Numbers?

if they have no positive common factor except 1 \hspace{0.2em} 1 \hspace{0.2em} 1 . For example, 4 \hspace{0.2em} 4 \hspace{0.2em} 4 and 5 \hspace{0.2em} 5 \hspace{0.2em} 5 are co-prime numbers. There is no positive integer other than 1 \hspace{0.2em} 1 \hspace{0.2em} 1 that divides into both 4 \hspace{0.2em} 4 \hspace{0.2em} 4 and 5 \hspace{0.2em} 5 \hspace{0.2em} 5 evenly.

Another example of co-prime numbers is 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 5 \hspace{0.2em} 5 \hspace{0.2em} 5 , and 9 \hspace{0.2em} 9 \hspace{0.2em} 9 .

12 \hspace{0.2em} 12 \hspace{0.2em} 12 and 18 \hspace{0.2em} 18 \hspace{0.2em} 18 are not co-prime because they have common factors other than 1 \hspace{0.2em} 1 \hspace{0.2em} 1 ( 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , 3 \hspace{0.2em} 3 \hspace{0.2em} 3 , and 6 \hspace{0.2em} 6 \hspace{0.2em} 6 ).

So, the greatest common factor (GCF) of co-prime numbers is 1 \hspace{0.2em} 1 \hspace{0.2em} 1 .

Co-prime numbers are also said to be “relatively prime” or “mutually prime”.

And with that, we come to the end of this tutorial. Until next time.

Frequently Asked Questions

Is 1 a prime number?

No. 1 is neither prime nor composite. By definition, a number has to be greater than 1 to be prime or composite.

Is 2 a prime number? And why?

Yes. 2 is a counting number and has exactly two positive factors – 1 and 2 itself. So it is a prime number. It is the smallest and the only even prime number.

Are all prime numbers odd?

No. 2 is an even prime number. But yes, with the exception of 2 , all prime numbers are odd.

An even number greater than 2 cannot prime because it will have at least three factors – 1 , the number itself, and 2 .

Are all odd numbers prime?

No. Most odd numbers are not prime. For example, 1 , 9 , 15 , 21 , 25 , 27 , etc.

What makes prime numbers so important?

Every natural number greater than 1 is either a prime itself or can be expressed (in a unique way) as a product of primes. For example, 6 can be written as 2 x 3 .

But you can’t resolve prime numbers into smaller factors. This makes prime numbers the building blocks of all counting numbers larger than 1 .

There’s more. For example, prime numbers are used extensively in cryptography .

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Prime Numbers: What and Why

I’ll begin a short series of posts on prime numbers with several questions on the basics: What are prime (and composite) numbers, and why do they matter?

What is a prime number?

We’ll start with an anonymous question from 1995:

Doctor Ken answered with the definition and a pair of examples:

As we’ll be seeing in the next two weeks, there are many ways to misstate this definition; every word in the definition matters!

This is where the word “exactly” comes in. Without it, you could take “has two factors, 1 and itself” as being true for 1, since 1 and itself are both factors and there is no other. (We’ll be seeing next week why we don’t want 1 to be a prime number; and the following week, we’ll see why “positive” matters.)

One important feature of prime numbers is that they are hard to predict, and seem almost random – yet they are definitely not! Here are the primes less than 100 on a number line:

Here are all the primes listed above, namely those less than 200:

The farther out you look, the more random they appear. Even here, you can see runs of nearly consecutive primes, and gaps with none.

How can I make a list like that?

Here is a 1997 question:

Doctor Wilkinson answered, starting with the basics:

This version of the definition is less formal, but gives the main idea well. We can directly use it to find primes (though not very efficiently).

One way to improve efficiency would be to try only prime divisors, so we’d skip 4. Why? Because if a number can be divided evenly by 4, then it would have already have been found to be divisible by 2. But we aren’t looking for the best possible way to accomplish the task; we just want Leah to experience what primes are – and maybe discover more about them by doing things that aren’t necessary.

This kind of thinking allows you to decide when to stop: If the quotient is smaller than the divisor, you don’t need to try any larger divisors.

This suggests other ways to shorten the work, which you can discover as you work. Later, we’ll look into how to test a number to see if it’s prime, and how to make a list more efficiently. But that can wait.

Prime and composite numbers

For more details, here is a question from 2003:

Doctor Ian answered:

We could also use $4\times3$, $6\times2$, and $12\times1$, with the same pairs in the reverse order; we’re interested only in the pairs, not the order.

It’s easy to see that prime numbers are special. They can’t be broken down, sort of like atoms in chemistry.

So composite numbers are numbers that are composed of other numbers. (And in chemistry, a compound is composed of different atoms! That’s related, too.)

On the other hand, prime means “ first “, or “most important”; primes are the numbers we start with when we build up other numbers by multiplication – the building blocks of the natural numbers.

So when we say that a prime number is one that has exactly two factors (itself and 1), we are saying that there are no “non-trivial” ways to factor it; it can’t be made by “putting together” numbers not including itself .

We say that the terms “prime” and “composite” are not exhaustive; “composite” doesn’t quite mean “not prime”. More on this, again, next time.

Who uses primes?

That was a reference to the following question from 2001:

Doctor Ian had answered that:

Encryption is everywhere now! And the basic idea behind the common method is that it’s easy to “compose” numbers, but hard to “decompose” them into a product of primes.

For example, say I choose the primes 7127 and 7879. Their product is 56,153,633. I send you this number (or just post it on my website for anyone to use); you use that number, by a method I’ve specified, to encrypt a message; and I then use my separate primes to create a number I can use to decrypt the message. This calculation is easy using the primes themselves, but would be very hard using only their product. On the other hand, anyone who could factor 56,153,633 could decrypt the message; so I’m trusting that it’s too hard for anyone to do quickly enough to take advantage of it. (We’d really use primes with 100 or more digits.)

But that’s all behind the scenes, and you don’t have to know about that math in order to use it. ( Somebody does have to know it, though!) How might primes be needed directly in your own experience? Mostly as a part of other math:

Knowing how a number breaks down, like knowing what atoms a chemical is made of, or how the parts of a car fit together, makes it possible to understand it better.

In the example above, you can list the pairs of factors by listing all ways to choose 0, 1, 2, or 3 twos, 0, 1, or 2 threes, and 0 or 1 five to make a factor. This tells you, in fact, that there are $4\times3\times2=24$ factors of 360 (that is, 12 pairs of factors). The 12 factors listed above are only half of them. (Can you find the other 12?)

(See Counting Divisors of a Number for more on this.)

For example, if the denominators of two fractions are, say, 2205 and 2100, that is, $3^2\times5\times7^2$ and $2^2\times3\times5^2\times7$, we know that the least common denominator, in order to be a multiple of both, has to have 2 twos, 2 threes, 2 fives, and 2 sevens, making it $2^2\times3^2\times5^2\times7^2=44,100$. We can find the greatest common factor similarly. Of course, since it is hard to find factors of large numbers, another method is needed when the numbers are really large.

(See Many Ways to Find the Least Common Multiple for more on this. Also, compare Three Ways to Find the Greatest Common Factor . And don’t forget How Do You Simplify a Fraction? .)

Are prime numbers necessary ? Not really:

And that’s what primes do most: save time (sometimes centuries, for really big jobs).

It’s not how a number is written that matters

Let’s close with a 1998 question from an entirely different perspective:

Computers commonly use base 2 (binary) for internal storage of numbers, and represent them in hexadecimal (base 16) to print them out more compactly. Does that affect prime numbers? How can we recognize them when written in those forms?

Writing a number in a particular base means using place values that are powers of the base. For example, 21 in base 10 means $21_{10}=2\times10^1+1\times10^0=2\times10+1=21$, while 15 in hexadecimal (base 16) means $15_{16}=1\times16^1+5\times16^0=1\times16+5=21$. They’re different numerals for the same number.

Doctor Mike answered:

Whether you write 21 as $21_{10}$ or as $15_{16}$, it is still a composite number; in either base, it is $3\times7$. And whichever way we write 37, it still refers to the same number – even though we are so familiar with numbers like 25 that we automatically think of it as a square. (A similar issue arises with judgments of evenness and oddness; in an odd base, numbers that “look” even to our base-ten eyes may be odd!)

And if a number has no factors other than itself and 1 in base ten, that is still true when you write it in another base. It’s the number that counts, not the numeral (the representation of the number).

So, how do you recognize a prime in binary or hexadecimal? The same way, ultimately, as in decimal: Either do a lot of divisions to look for factors, or be sufficiently fluent in the appropriate base that you recognize products (or know the divisibility tests appropriate to that base – which will all be different than in base ten).

So a prime base has no effect on primality of numbers written in it, any more than any other base does. It just makes things look different.

Just for fun, here are the first 13 primes, written in base 13 (with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C):

2, 3, 5, 7, 11, 10, 14, 16, 1A, 23, 25, 2B, 32

They don’t all look prime to eyes accustomed to base ten, but they are! (Note that the units digit being even doesn’t imply the number is even, when the base is odd.)

Next week, we’ll look at some special cases: 0 and 1. The following week, we’ll consider negative numbers.

5 thoughts on “Prime Numbers: What and Why”

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i am looking for a list of prime numbers and each prime number gap that runs from 2 to up to say 5 millionth prime number. ( or a very large available prime such as one millionth prime ). all the google and any other internet search i have tried jumps over this direct listing of primes and their gaps so i am stopped. a few examples are: 2 1, 3 2,5 2,7 4 and on up to the largest available. can you help with this? thank you.

You could, of course, just take any list of 5 million primes, and write a simple program (or spreadsheet) to calculate the gaps.

Alternatively, the list of gaps is in OEIC here , and has a link to Vojtech Strnad, First 100000 terms [First 10000 terms from N. J. A. Sloane] . That doesn’t include the primes themselves, but you could correlate it with a list of primes.

How to Check if a Number Is Prime

Last Updated: February 28, 2024 Fact Checked

Prime Tests

Understanding prime testing, chinese remainder theorem test, things you'll need.

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 892,283 times.

Prime numbers are those divisible only by themselves and 1; all others are called composite numbers. While there are numerous ways to test for primality, there are trade offs. Perfect tests exist, but are extremely slow for large numbers, while much faster can give false results. Here are a few options to choose from depending on how large a number you are testing.

Note: In all formulas, n is the number being tested for primality.

Choose an integer value for a such that 2 ≤ a ≤ n - 1.
If a n (mod n) = a (mod n), then n is likely prime. If this is not true, n is not prime.
Repeat with different values of a to increase confidence in primality

Choose an integer value for a such that 2 ≤ a ≤ n - 1.
If a d = +1 (mod n) or -1 (mod n), then n is probably prime. Skip to test result. Otherwise, go to next step.

$a^{{2d}}$

Test result: If n passes test, repeat with different values of a to increase confidence.

Step 1 Understand the trial division method.

Floor(x) rounds x to the closest integer ≤ x.

Many calculators have a mod button, but see the end of this section for how to solve this by hand for large numbers.

Step 3 Know the pitfalls of Fermat's Little Theorem.

"Prime1" = 35
Prime2 = 97

Step 2 Choose two datapoints that are greater than zero and less than prime1 and prime2 respectfully.

MMI1 = Prime2 ^ -1 Mod Prime1
MMI2 = Prime1 ^ -1 Mod Prime2
MMI1 = (Prime2 ^ (Prime1-2)) % Prime1
MMI2 = (Prime1 ^ (Prime2-2)) % Prime2
MMI1 = (97 ^ 33) % 35
MMI2 = (35 ^ 95) % 97

Step 4 Create a binary table for each MMI up to Log2 of the Modulus

F(1) = Prime2 % Prime1 = 97 % 35 = 27
F(2) = F(1) * F(1) % Prime1 = 27 * 27 % 35 = 29
F(4) = F(2) * F(2) % Prime1 = 29 * 29 % 35 = 1
F(8) = F(4) * F(4) % Prime1 = 1 * 1 % 35 = 1
F(16) =F(8) * F(8) % Prime1 = 1 * 1 % 35 = 1
F(32) =F(16) * F(16) % Prime1 = 1 * 1 % 35 = 1
35 -2 = 33 (10001) base 2
MMI1 = F(33) = F(32) * F(1) mod 35
MMI1 = F(33) = 1 * 27 Mod 35
F(1) = Prime1 % Prime2 = 35 % 97 = 35
F(2) = F(1) * F(1) % Prime2 = 35 * 35 mod 97 = 61
F(4) = F(2) * F(2) % Prime2 = 61 * 61 mod 97 = 35
F(8) = F(4) * F(4) % Prime2 = 35 * 35 mod 97 = 61
F(16) = F(8) * F(8) % Prime2 = 61 * 61 mod 97 = 35
F(32) = F(16) * F(16) % Prime2 = 35 * 35 mod 97 = 61
F(64) = F(32) * F(32) % Prime2 = 61 * 61 mod 97 = 35
F(128) = F(64) * F(64) % Prime2 = 35 * 35 mod 97 = 61
97 - 2 = 95 = (1011111) base 2
MMI2 = (((((F(64) * F(16) % 97) * F(8) % 97) * F(4) % 97) * F(2) % 97) * F(1) % 97)
MMI2 = (((((35 * 35) %97) * 61) % 97) * 35 % 97) * 61 % 97) * 35 % 97)

Step 5 Calculate (Data1 * Prime2 * MMI1 + Data2 * Prime1 * MMI2) % (Prime1 * Prime2)

Answer = (1 * 97 * 27 + 2 * 35 * 61) % (97 * 35)
Answer = (2619 + 4270) % 3395
Answer = 99

Calculate (Answer - Data1) % Prime1
99 -1 % 35 = 28
Since 28 is greater than 0, 35 is not prime

Calculate (Answer - Data2) % Prime2
99 - 2 % 97 = 0
Since 0 equals 0, 97 is potentially prime

Step 8 Repeat steps 1 through 7 at least two more times.

Use a different "prime1" where prime1 is a non-prime
Use a different prime 1 where prime 1 is an actual prime. In this case, steps 6 and 7 should equal 0.
Use different data points for data1 and data2.
If step 7 is 0 every time, there is an extremely high probability that prime2 is prime.
Steps 1 though 7 are known to fail in certain cases when the first number is a non-prime number and the second prime is a factor of the non-prime number "prime1". It works in all scenarios where both numbers are prime.
The reason why steps 1 though 7 are repeated is because there are a few scenarios where, even if prime1 is not prime and prime2 is not prime, step 7 still works out to be zero, for one or both the numbers. These circumstances are rare. By changing prime1 to a different non-prime number, if prime2 is not prime, prime2 will rapidly not equal zero in step 7. Except for the instance where "prime1" is a factor of prime2, prime numbers will always equal zero in step 7.

Community Q&A

The 168 prime numbers under 1000 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 [10] X Research source Thanks Helpful 0 Not Helpful 0
While trial division is slower than more sophisticated methods for large numbers, it is still quite efficient for small numbers. Even for primality testing of large numbers, it is not uncommon to first check for small factors before switching to a more advanced method when no small factors are found. Thanks Helpful 0 Not Helpful 0

Working out tools, such as paper and pen or a computer

↑ https://primes.utm.edu/glossary/page.php?sort=TrialDivision
↑ https://mathworld.wolfram.com/FermatsLittleTheorem.html
↑ https://crypto.stanford.edu/pbc/notes/numbertheory/millerrabin.html
↑ http://betterexplained.com/articles/fun-with-modular-arithmetic/
↑ http://mathworld.wolfram.com/FermatsLittleTheorem.html
↑ http://www.cs.cornell.edu/courses/cs4820/2010sp/handouts/MillerRabin.pdf
↑ https://faculty.math.illinois.edu/~dsamart/PrimalityTests.pdf
↑ https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/modular-exponentiation
↑ Online Encyclopedia of Integer Sequences, A000040
Topcoder.com - sample source code and documentation for methods discussed here
Online Prime Number Checker - check numbers with up to 5000 digits

About This Article

To check if a number is prime, divide it by every prime number starting with 2, and ending when the square of the prime number is greater than the number you’re checking against. If it is not evenly divided by any whole number other than 1 or itself, the number is prime. If you want to learn how to do modular arithmetic to test large numbers, keep reading the article! Did this summary help you? Yes No

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Prime Numbers

Primality test algorithms.

Introduction to Primality Test and School Method
Fermat Method of Primality Test
Primality Test | Set 3 (Miller–Rabin)
Solovay-Strassen method of Primality Test
Lucas Primality Test
How is the time complexity of Sieve of Eratosthenes is n*log(log(n))?
Sieve of Eratosthenes in 0(n) time complexity

Programs and Problems based on Sieve of Eratosthenes

C++ Program for Sieve of Eratosthenes
Java Program for Sieve of Eratosthenes
Scala | Sieve of Eratosthenes
Check if a number is Primorial Prime or not
Sum of all Primes in a given range using Sieve of Eratosthenes
Prime Factorization using Sieve O(log n) for multiple queries
Java Program to Implement Sieve of Eratosthenes to Generate Prime Numbers Between Given Range
Segmented Sieve
Segmented Sieve (Print Primes in a Range)
Longest sub-array of Prime Numbers using Segmented Sieve
Sieve of Sundaram to print all primes smaller than n
Bitwise Sieve
Sieve of Atkin
Wheel Factorization Algorithm
Java Program to Implement wheel Sieve to Generate Prime Numbers Between Given Range

What are Prime Numbers?

A prime number is defined as a natural number greater than 1 and is divisible by only 1 and itself.

In other words, the prime number is a positive integer greater than 1 that has exactly two factors, 1 and the number itself. First few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 . . .

Note: 1 is not either prime or composite. The remaining numbers, except for 1, are classified as prime and composite numbers.

Prime numbers

Some interesting facts about Prime Numbers:

Except for 2, which is the smallest prime number and the only even prime number, all prime numbers are odd numbers.
Every prime number can be represented in form of 6n + 1 or 6n – 1 except the prime numbers 2 and 3 , where n is any natural number.
2 and 3 are only two consecutive natural numbers that are prime.
Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
Wilson Theorem : Wilson’s theorem states that a natural number p > 1 is a prime number if and only if

(p – 1) ! ≡ -1 mod p OR, (p – 1) ! ≡ (p-1) mod p

Fermat’s Little Theorem : If n is a prime number, then for every a, 1 ≤ a < n,

a n-1 ≡ 1 (mod n) OR, a n-1 % n = 1

Prime Number Theorem : The probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
Lemoine’s Conjecture : Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture.

Properties of Prime Numbers:

Every number greater than 1 can be divided by at least one prime number.
Every even positive integer greater than 2 can be expressed as the sum of two primes.
Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
Two prime numbers are always coprime to each other.
Each composite number can be factored into prime factors and individually all of these are unique in nature.

Prime Numbers and Co-prime numbers:

It is important to distinguish between prime numbers and co-prime numbers . Listed below are the differences between prime and co-prime numbers.

Coprime numbers are always considered as a pair, whereas a prime number is a single number.
Co-prime numbers are numbers that have no common factor except 1. In contrast, prime numbers do not have such a condition.
A co-prime number can be either prime or composite, but its greatest common factor (GCF) must always be 1. Unlike composite numbers, prime numbers have only two factors, 1 and the number itself.
Example of co-prime: 13 and 15 are co-primes. The factors of 13 are 1 and 13 and the factors of 15 are 1, 3 and 5. We can see that they have only 1 as their common factor, therefore, they are coprime numbers.
Example of prime: A few examples of prime numbers are 2, 3, 5, 7 and 11 etc.

How to check whether a number is Prime or not?

Naive Approach: The naive approach is to

Iterate from 2 to (n-1) and check if any number in this range divides n . If the number divides n , then it is not a prime number.

Time Complexity: O(N) Auxiliary Space: O(1)

Naive approach (recursive): Recursion can also be used to check if a number between 2 to n – 1 divides n. If we find any number that divides, we return false.

Below is the implementation of the above idea:

Time Complexity: O(N) Auxiliary Space: O(N) if we consider the recursion stack. Otherwise, it is O(1).

Efficient Approach: An efficient solution is to:

Iterate through all numbers from 2 to ssquare root of n and for every number check if it divides n [because if a number is expressed as n = xy and any of the x or y is greater than the root of n, the other must be less than the root value]. If we find any number that divides, we return false.

Below is the implementation:

Time Complexity: O(sqrt(n)) Auxiliary space: O(1)

Another Efficient approach: To check whether the number is prime or not follow the below idea:

We will deal with a few numbers such as 1, 2, 3, and the numbers which are divisible by 2 and 3 in separate cases and for remaining numbers. Iterate from 5 to sqrt(n) and check for each iteration whether (that value) or (that value + 2) divides n or not and increment the value by 6 [because any prime can be expressed as 6n+1 or 6n-1]. If we find any number that divides, we return false.

Below is the implementation for the above idea:

Time complexity: O(sqrt(n)) Auxiliary space: O(1)

Efficient solutions

Primality Test | Set 1 (Introduction and School Method)
Primality Test | Set 2 (Fermat Method)
Primality Test | Set 4 (Solovay-Strassen)

Algorithms to find all prime numbers smaller than the N.

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Prime Numbers From 1 to 1000

Prime Numbers from 1 to 1000

Prime numbers from 1 to 1000 will include the list of primes, that have only two factors, i.e. 1 and the number itself. To find the prime numbers from 1 to 1000, we need to check if the number is a natural number and has no positive divisor other than 1 and itself. We do not consider 1 as a prime number , as it has only one factor but other prime numbers have two factors.

For example, 5 is a prime number, because it has only two factors, 1 and 5, such as;

But 4 is not a prime number, as it has more than two factors, 1, 2, and 4, such as,

Here, 4 is said to be a composite number . These factors can be determined with the help of the prime factorisation method.

List of Prime Numbers 1 to 1000

Now, let us see here the list of prime numbers starting from 1 to 1000. We should remember that 1 is not a prime number, as it has only one factor. Thus, the prime numbers start from 2.

From the above list of prime numbers, we can find that each of the primes has only two factors.

Prime Numbers 1 to 1000 – Download PDF

How to find prime numbers from 1 to 1000.

By the definition of prime numbers, we know that the prime number will have only two factors. In the above-given list, the numbers provided are all prime numbers. We can cross-check with any of these numbers to know if they are prime or not, by prime factorising them.

For example:

709 = 1 x 709, only two factors
911 = 1 x 911, only two factors
401 = 1 x 401, only two factors

But if we check with any other, let us say, 15, then the factors of 15 are:

1 x 15 = 15

Thus, there is a total of four factors: 1, 3, 5, and 15. So, 15 is not a prime number.

Therefore, this way we can find all the prime numbers. Let us see some of the properties of prime numbers, to make it easier to find them.

Properties of Prime Numbers

A prime number will have only two factors, 1 and the number itself
2 is the only even prime number
All the even numbers greater than 2, are the product of two or more prime numbers

Video Lesson on Prime Numbers

Prime Numbers Up to 100
Prime Factorization
Prime Factors
Co-Prime Numbers
How to Find Prime Numbers?

Solved Examples

Q.1: From the list of prime numbers 1 to 1000 given above, find if 825 is a prime number or not?

Solution: The list of prime numbers from 1 to 1000 does not include 825 as a prime number. It is a composite number since it has more than two factors. We can confirm this by prime factorisation of 825 also.

Prime Factorization of 825 = 3 1 × 5 2 × 11 1

Hence, 825 includes more than two factors.

Q.2: Find if 911 is a prime number or not?

Solution: Since, 911 has only two factors, 1 and 911. Therefore, 911 is a prime number.

911 = 1 x 911

Q.3: What are the factors of 443?

Solution: Since 443 is a prime number, therefore, it has only two factors 1 and 443.

Q.4: What are the prime numbers from 1 to 300? How many are they?

Solution: The prime numbers from 1 to 300 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293.

Therefore, there is a total of 62 prime numbers between 1 to 300.

Practice Questions

1 is a prime number – ______
2 is the only even prime number – ______
1000 is not a prime number – ______
A prime number has more than two factors – ______
Find if 221 is a prime number or not.

Frequently Asked Questions on Prime Numbers 1 to 1000

How many prime numbers are there in between 1 to 1000, what are the prime numbers from 1 to 200, how to find if a number is prime, why 11 is a prime number.

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Prime Video Now Reaches More Than 200 Million Monthly Viewers, TV Ads ‘Off to a Strong Start,’ Amazon CEO Says

By Todd Spangler

Todd Spangler

NY Digital Editor

Chicken Soup for the Soul Entertainment Sued Over Redbox Acquisition by Consultant Who Claims He’s Owed Several Million Dollars 6 hours ago
Netflix Rattles Investors by Ending Subscriber Disclosures — but Apple’s Similar Strategy in 2018 With iPhones Was a Big Success 12 hours ago
Watcher Entertainment Launches Its Own Subscription Streaming Service: ‘We’re Leaving YouTube’ 16 hours ago

Prime Video has a big audience, and now Amazon CEO Andy Jassy has put a new number on it: The premium video service has more than 200 million monthly viewers.

Jassy revealed the number in Amazon’s annual letter to shareholders Thursday. “Recently, we’ve expanded our streaming TV advertising by introducing ads into Prime Video shows and movies, where brands can reach over 200 million monthly viewers in our most popular entertainment offerings,” the CEO wrote, spanning movies, TV shows and live sports like the NFL’s “Thursday Night Football.”

Popular on Variety

Note that “viewers” is a different metric than subscribers (or accounts). For example, as of the end of 2023, Netflix counted 260.28 million total subscribers globally, meaning Netflix’s reach in terms of total number of viewers is even bigger than that.

In the letter, Jassy also reiterated that Amazon believes Prime Video — which started as a way to enhance the value of the Prime free-shipping program and to drive up ecommerce purchases — can be a lucrative standalone business .

“We have increasing conviction that Prime Video can be a large and profitable business on its own,” the CEO wrote. “This confidence is buoyed by the continued development of compelling, exclusive content (e.g. Thursday Night Football, Lord of the Rings, Reacher, The Boys, Citadel, Road House, etc.), Prime Video customers’ engagement with this content, growth in our marketplace programs (through our third-party Channels program, as well as the broad selection of shows and movies customers rent or buy), and the addition of advertising in Prime Video.”

Also in Jassy’s letter to shareholders, he touted the promise of generative AI, saying it “may be the largest technology transformation since the cloud (which itself, is still in the early stages), and perhaps since the internet… [T]his GenAI revolution will be built from the start on top of the cloud. The amount of societal and business benefit from the solutions that will be possible will astound us all.”

Amazon has invested $4 billion in AI start-up Anthropic for a minority stake in the company. Under the partnership, Anthropic is using Amazon’s custom chips to build, train and deploy its future models and has made a long-term commitment to provide AWS customers access “to future generations of its foundation models.”

“We’re building a substantial number of GenAI applications across every Amazon consumer business,” although “the vast majority will ultimately be built by other companies,” Jassy wrote in the letter. Amazon’s projects in this area include the AI-powered shopping assistant Rufus, an “even more intelligent and capable” Alexa and advertising capabilities to make it “simple with natural-language prompts to generate, customize and edit high-quality images, advertising copy and videos,” he wrote.

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‘I’m Just Going to Vibe With It’: Nelly Korda One Shot Back at Chevron Championship

Korda enters the week on a four-event winning streak, and she’s in prime position at the first women’s major of the year.

Author: Jeff Ritter

With all due respect to recent Masters champion Scottie Scheffler , the hottest player in golf is competing just outside Houston this weekend.

And through 36 holes, she’s in position to ring up a fifth straight victory.

Scheffler has been incredible on the PGA Tour, but on the LPGA no one has beaten Nelly Korda for four consecutive events . On Friday at the Chevron Championship, the first women’s major of the year, Korda didn’t play her cleanest round but still shot a 3-under 69 to move to 7 under for the event, leaving her in prime position heading into the weekend: one shot back.

“It takes a lot of patience to win. At the end of the day the person that makes the least amount of mistakes or recovers the best from their mistakes ends up usually winning,” she said.

Korda actually opened her second round with a double bogey that would shake the confidence of a player who doesn’t happen to possess supreme confidence entering that moment. Korda, as one might expect, was unfazed.

“I actually didn't feel bad at all,” she said of her opening double. “Sometimes when you start to make mistakes you just don't really feel confident or you don't feel that great. But I just kind of told myself that it's the first hole of the tournament today. Even though I may have made a double, I wanted to save a bogey. There is still so much golf to be played and there is still a good bit of gettable par-5s.

“So that's usually what I think about, is just the opportunities that I have ahead.”

After her opening wobble, Korda made birdies on 2 and 4, and after a bogey on 7 she bounced back with birdies on 8 and 9. She went bogey-free on the back nine thanks in part to a nice par escape on 14, when she drove it into thick grass right of the fairway, hacked out and drained a 12-footer that took a 360-degree trip around the rim before dropping.

. @NellyKorda absolutely grinding it out for the par-save She maintains her co-lead position at -6 💪 pic.twitter.com/q3cwSaR1Yy — LPGA (@LPGA) April 19, 2024

Korda is the first American to win four straight LPGA events since Nancy Lopez won five in a row during her nine-win rookie season in 1978.

Jin Hee Im and Atthaya Thitikul share the 36-hole lead at 8 under. Hae Ran Ryu is 6 under and solo fourth, two back of the lead and one behind Korda. Thitikul could be the most formidable of that group, as she entered the week No. 10 in the women’s world rankings and was the 2022 Rookie of the Year.

But for now all eyes remain on Korda as she chases he 13th career title, fifth in a row and second major.

"I'm just at the halfway point right now. The amount of golf that I've played, I still have that to go. There is still a lot of golf left and anything can happen,” she said.

“Just going to stick to my process and vibe with it, is what my coach says.”

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Masters a reunion of the world’s best players. But the numbers are shrinking

Tyrrell Hatton, of England, reacts on the 11th hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Ashley Landis)

Phil Mickelson warms up on the driving range before a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Charlie Riedel)

Scottie Scheffler hits his tee shot on the 10th hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Charlie Riedel)

Sam Burns waits to hit on the 10th hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Matt Slocum)

Hideki Matsuyama, of Japan, putts on the 11th hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Matt Slocum)

Jon Rahm, of Spain, watches his tee shot on the fourth hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/George Walker IV)

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AUGUSTA, Ga. (AP) — More than golf’s first major championship of the year, the Masters represents unification. This is the first time since July at the British Open the best players regardless of their tours compete against each other — same course, same tournament, same television network.

“I believe everyone agrees there’s excitement in the air this week,” Masters Chairman Fred Ridley said Wednesday. “The best players in the world are together once again.”

Still unclear at Augusta National is for how much longer.

Saudi-funded LIV Golf has 13 players at the Masters, seven of them former champions who can play as long as they want. That’s down from 18 a year ago. Only nine LIV players are assured of being back to Augusta National next year, depending on how they fare in the majors this year.

Ridley offered little hope the pathway for LIV to Augusta National was about to get wider.

He said the Official World Golf Ranking was a “legitimate determiner” of the best in golf, bad news for a rival league that does not get world ranking points. And while the Masters annually reviews its criteria for invitations, Ridley announced no new changes.

Scottie Scheffler chips to the green on the 17th hole during the second round of the RBC Heritage golf tournament, Friday, April 19, 2024, in Hilton Head Island, S.C. (AP Photo/Chris Carlson)

Instead, he leaned on the Masters being an invitational, and the club alone decides who it deems worthy of getting that elegant, cream-colored invitation in the mail.

“If we felt that there were a player or players, whether they played on the LIV Tour or any other tour, who were deserving of an invitation to the Masters, we would exercise that discretion with regard to special invitations,” Ridley said.

The battle is for a green jacket, but that might not be the only competition.

It will be difficult to look at a leaderboard without considering who is with LIV Golf. That much hasn’t changed from last year — the first Masters since LIV was launched — and LIV certainly showed the 54-hole, no-cut league didn’t affect them. Three players were among the top four on the final leaderboard.

And just like last year, there is no animosity inside the ropes.

Phil Mickelson and Joaquin Niemann from LIV Golf played a practice round with Akshay Bhatia, the final player into the field because of his Texas Open victory last week. Xander Schauffele told of running into Dustin Johnson and the two decided to play a practice round, no different from what would have happened long before LIV began luring away players with guaranteed riches.

But the future remains murky.

Augusta National and the other three organizations that run majors have seats on the OWGR board that reviewed LIV’s application to join and get world ranking points. The vote was unanimous not to award points until certain enhancements were met.

LIV eventually decided to withdraw its application , and several players decried the world ranking as no longer relevant.

It is to Ridley and the Masters. The top 50 at the end of the year and a week before the Masters still get invitations. Bryson DeChambeau said the majors, including the Masters, should invite the top 12 from the LIV points list.

Ridley wasn’t buying that.

Jon Rahm, of Spain, watches his tee shot on the fourth hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/George Walker IV)

“I think it will be difficult to establish any type of point system that had any connection to the rest of the world of golf because they’re basically — not totally, but for the most part — a closed shop,” Ridley said. “There is some relegation, but not very much.

“But I don’t think that prevents us from giving subjective consideration based on talent, based on performance to those players.”

That’s what led Augusta National to offer an invitation to Niemann. The club did not cite anything he did on LIV — the Chilean has two LIV wins this year — but his willingness to travel outside LIV and win the Australian Open, along with a top finish in the Australian PGA.

Talor Gooch did not get an invitation. He won three LIV events last year and later suggested Rory McIlroy would have an asterisk next to his name if he won the Masters because all the best aren’t there.

Gooch is unlikely to be missed, not with Scottie Scheffler going for a second green jacket, with McIlroy chasing the career Grand Slam, Tiger Woods playing for only the second time this year and a host of others from all tours chasing one of golf’s most prized possessions.

And then the PGA Tour will head to Hilton Head and LIV Golf will make its way to Australia, and they all have to wait until the next major May 16-19 at the PGA Championship.

“There’s a lot of people a lot smarter than me that could figure this out in a much more efficient way,” Jon Rahm said. “But the obvious answer is that there’s got to be a way for certain players in whatever tour to be able to earn their way in. That’s the only thing can I say. I don’t know what that looks like. But there’s got to be a fair way for everybody to compete.”

AP golf: https://apnews.com/hub/golf

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Prime Numbers
A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself. For example, 5 is a prime number because it has no positive divisors other than 1 and 5. In contrast to prime numbers, a composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 4 is a composite number because it has three positive ...
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Find out if a number is Prime or not (works on numbers up to 4,294,967,295): You can also try this Prime Numbers Activity . Prime and Composite Numbers Prime Factorization Tool Coprime Calculator Prime Properties Prime Numbers - Advanced Prime Number Lists.
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Example 2: Express 21 as the sum of two prime numbers. Solution: We can write 21 as. $21 = 19 + 2$ Here, 2 and 19 are both prime numbers. Example 3: What prime numbers are there between 20 and 30? Solution: The prime numbers between 20 and 30 are 23 and 29. Example 4: What is the greatest prime number between 80 and 90? Solution:
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
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Well, you cannot find every prime number, because Euclid proved sometime around 300 BC that there are an infinite number of them. If you want to find all prime numbers up to some given limit, n, you may resort to the algorithm known as the Sieve of Eratosthenes: Write down all numbers from 2 to n; Start with the smallest number in our list: 2.Circle 2 and cross out all consecutive multiples of ...
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History of Prime Numbers. The prime number was discovered by Eratosthenes (275-194 B.C., Greece). He took the example of a sieve to filter out the prime numbers from a list of natural numbers and drain out the composite numbers.. Students can practise this method by writing the positive integers from 1 to 100, circling the prime numbers, and putting a cross mark on composites.
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A prime number is divisible only by itself and 1. There are 25 prime number less than 100. A prime number is a natural number that can only be divided, without a remainder, by itself and 1. In other words, a prime number has exactly two factors. For example, 13 is only divisible by 13 and 1. In contrast, a composite number is a natural number ...
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Let f be an integer-coefficient irreducible polynomial with degree higher than 2, and let k = gcd(f(0), f(1)). The conjecture: f(n) / k always generates an infinite number of primes. Some polynomials, like x12 + 488669 seem to only sparsely make prime numbers, but so far no bounds are known for any of these polynomials. Share.
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Masters a reunion of the world's best players. But the numbers are shrinking. Tyrrell Hatton, of England, reacts on the 11th hole during a practice round in preparation for the Masters golf tournament at Augusta National Golf Club Wednesday, April 10, 2024, in Augusta, GA. (AP Photo/Ashley Landis)

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