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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.

## Related Posts

- 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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- Quiz on Fibonacci Numbers

## 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.

- Sieve of Sundaram
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- Program to print all prime factors of a given number
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- Prime factors of LCM of array elements – GeeksforGeeks
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## 'The highest numbers I ever did' – Tadej Pogačar in dominant position at Tour de France after record display

Slovenian expects Visma onslaught in Alps, hints at riding Vuelta in 2025

The performance may have exceeded any ever produced in the mountains of the Tour de France , but the aftermath was familiar. At this race, stories have a habit of repeating themselves through the eras as though they had never happened before.

When Tadej Pogačar produced a display for the ages at Plateau de Beille on Sunday afternoon, breaking the late Marco Pantani's record for the ascent by an estimated 3:40, he was always likely to face a familiar question in his rest day press conference: how could he explain it?

These days, a winning rider will rarely say that they were simply much stronger than everybody else. Pogačar gave much of the credit for his remarkable time at Plateau de Beille to the pace-making of Jonas Vingegaard and his Visma-Lease a Bike teammate Matteo Jorgenson on the lower slopes.

"Yesterday, I think we all witnessed one of the best performances on the climbs ever," Pogačar told reporters in a video conference on Monday afternoon. "For myself also, when I checked my numbers afterwards, it was really crazy, especially the part where Matteo Jorgenson and Jonas went on the front – that was the highest numbers I ever did in my career. It was a big day.

"We can see that Jonas came here really prepared to fight for the victory. Yesterday, they really showed the balls, finally, and they hit hard. So really hats off to them for yesterday, for the whole Visma team, they did a super good ride. In the end, it was just an all-out effort from the bottom to the top. It was crazy, crazy stage."

Pogačar's performance did not take place in a vacuum. The first three riders on the stage all finished inside Pantani's 1998 time, after all, while this season alone has seen record-setting average speeds at six Classics, including Milan-San Remo, the Tour of Flanders and Paris-Roubaix.

Since the turn of the decade, the sheer intensity of bike racing has morphed almost beyond all recognition, and that process has seemed to accelerate still further in 2024. The more cautious tactical approaches of yesteryear have been overtaken by the almost reckless abandon of a select cadre of today's peloton.

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"Cycling is evolving so much, I must say," Pogačar said. "Six years ago, when I came to this team, and I don't want to speak bad about the team, it was totally different. If I compare this year to my first year in Vuelta, it was almost amateur. Back then, I thought everything was professional, but we moved on really fast. Every team pushes each other with technology, with nutrition, with training plans, with altitude camps. Especially Visma and UAE, and Ineos and Trek and QuickStep, we push each other to reach new limits.

"Yesterday, we witnessed the fastest-ever climbing and we should be seeing something like this every year because everybody is focusing so much on the details – every single gram of the food, every single watt you can save on the bike. We are going super fast. For me, it's really impressive to see how things changed in the last six years of my professional career."

On Friday, Escape Collective reported that UAE Team Emirates, Visma-Lease a Bike and Israel-Premier Tech have been using carbon monoxide rebreathers to optimise the benefits of altitude training. When Pogačar was asked on Monday to give some specific examples of what had changed since his professional career began in 2019, he reached for a rather humbler innovation.

"For example, six years ago, when I started, it was a lot about carbohydrates, we had white pasta, white rice and maybe omelette for breakfast," he said. "Now we have more normal breakfast, like rice porridge, oatmeal, pancakes, bread. I think this little thing already makes a difference.

"The bikes are also so much faster, especially the tyres. The tyres make the biggest difference from what we had six years ago or 10 years ago, and the wheels, aerodynamics, frames, it's just amazing how different the bike is now."

## The week ahead

In the Tour de France overall standings , Pogačar carries a lead of 3:09 over Vingegaard into the final six stages of the race, with Remco Evenepoel, third at 5:19, the only other rider within ten minutes of his maillot jaune. On the weekend's evidence, Pogačar is strongly favoured to claim his third Tour victory, but he knows that Vingegaard and his Visma squad are unlikely to lay down arms just yet.

The final week includes a tricky run to Superdévoluy on Wednesday and then back-to-back summit finishes at Isola 2000 and the Col de Couillole ahead of the final time trial to Nice. Pogačar earmarked those final two road stages in the Alps as the days where Vingegaard and Visma were likely to go for broke.

"I don't think they're picking both stages, Friday and Saturday, I think they will focus on one of them," Pogačar said. "We will try to do our own race and try to defend so they cannot do anything crazy. For sure they will try.

"Jonas said yesterday that he is not giving up on the fight, and I think that's correct what he said. It's going to be a tough final week and we're going to see a lot of fireworks for sure, from everybody."

At this remove, Pogačar looks destined to ride into Nice in yellow and become the first man since Pantani in 1998 to win the Giro-Tour double, but he – all but – dismissed the prospect of adding the Vuelta a España to his schedule for a tilt at the treble.

"This year it's 99% impossible," he said. "For next year, it's a much bigger chance to see me in the Vuelta."

Get unlimited access to all of our coverage of the Tour de France - including breaking news and analysis reported by our journalists on the ground from every stage of the race as it happens and more. Find out more .

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Barry Ryan is Head of Features at Cyclingnews. He has covered professional cycling since 2010, reporting from the Tour de France, Giro d’Italia and events from Argentina to Japan. His writing has appeared in The Independent, Procycling and Cycling Plus. He is the author of The Ascent: Sean Kelly, Stephen Roche and the Rise of Irish Cycling’s Golden Generation , published by Gill Books.

'We gambled and lost but no regrets' – Remco Evenepoel outgunned on final Tour de France summit finish

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## Most Popular

## Missed Prime Day? You can still save 20% on the number one best-selling air purifier on Amazon

The Levoit Core 300S is a #1 bestseller on Amazon ― and now you can snap up this top-rated air purifier with a 20% discount from a brand we know and trust.

Looking for a high-performing air purifier that can easily fit into a small bedroom? Then you can't miss this deal ― the bestselling Levoit Core 300S-P is now 20% off at Amazon , saving you a neat $30.

We are big fans of Levoit here at Live Science having reviewed their Levoit 300S humidifier , 400S air purifier and Vital 200S . Home appliances from this brand take pride of place in our guides to the best air purifiers and best air purifiers for allergies , with most models getting a near-perfect score of 4.5 out of 5 stars in our reviews. We like how sleek, quiet and easy to use they are, but more importantly, we are impressed by their air-cleaning performance given that they are relatively affordable. But do not take our word for it. Levoit air purifiers are one of the most sought-after appliances on Amazon, with the Levoit Core 300S-P being the number-one bestseller on the platform.

With its powerful three-stage filtration system and patented VortexAir technology, this compact air purifier can remove a whopping 99.97% of ultra-fine particles, including common allergens like pollen, dust and pet dander. Thanks to its activated carbon filter, it will do a good job of getting rid of bad smells, too. The Levoit Core 300S-P is also smart ― it comes with a free VeSync app that allows you to monitor your air quality and set schedules, and it can be controlled with Alexa or Google voice assistant. Moreover, this air purifier is Energy Star certified so it will not put a strain on your energy bill. It is basically a smaller version of our favorite Levoit Core 600S , making it a perfect option for bedrooms measuring up to 219 square feet (20 square meters).

For just $119.99, the Levoit Core 300S-P is an absolute steal.

Levoit Core 300S-P: Was $149.99, now $119.99 at Amazon

Save $30 on one of the most popular air purifiers at Amazon. Compact, quiet and smart-enabled, the Levoit Core 300S-P is a perfect choice for small- to medium-sized bedrooms.

There are two more models in the Levoit Core 300 family. The Levoit Core P350 comes with an enhanced activated carbon filter and is designed primarily with pet owners in mind. The Levoit Core 300-P, on the other hand, is the same as the Core 300S-P, but without the smart-enabled features. Both of them are on offer, too. The Levoit Core P350 is now 20% off , bringing its usual price of $129.99 down to $103.99, while the Levoit Core 300-P in white is now 21% cheaper , slashing its price tag by $21.

Key features: Pre-filter, multi-layer HEPA filter, activated carbon filter, air quality indicator, timer, auto mode, sleep mode, smart controls.

Product launched: September 2019

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Price history: For the better part of this year, the price of the Levoit Core 300S-P sat at $149.99. Today's offer from Amazon brings the price down to $119.99, which is the lowest price we have seen since Black Friday when it briefly dipped to $104.99.

Price comparison: Levoit: $129.99

Reviews consensus: The Levoit Core 300S punches way above its affordable price point. This sleek air purifier is compact, quiet, intuitive to use and packed with smart features, and it delivers great air-cleaning performance. It has several minor flaws, including delays in app response time, and it is not suitable for larger bedrooms — but ultimately, the Levoit Core 300S offers a great run for the money.

TechRadar: ★★★★ ½

✅ Buy it if: You are looking for a quiet and efficient air purifier for small to medium spaces.

❌ Don't buy it if: You want something bigger and more powerful.

Check out our other guides to the best air purifiers , air purifiers for allergies , the best telescopes , microscopes for students , binoculars , rowing machines , electric toothbrushes and more.

Anna Gora is a health writer at Live Science, having previously worked across Coach, Fit&Well, T3, TechRadar and Tom's Guide. She is a certified personal trainer, nutritionist and health coach with nearly 10 years of professional experience. Anna holds a Bachelor's degree in Nutrition from the Warsaw University of Life Sciences, a Master’s degree in Nutrition, Physical Activity & Public Health from the University of Bristol, as well as various health coaching certificates. She is passionate about empowering people to live a healthy lifestyle and promoting the benefits of a plant-based diet.

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## Amazon Says Prime Day 2024 Set Records

Key takeaways.

- Amazon said this year's Prime Day produced records for sales and items sold, though it didn't provide numbers.
- An analysis from Adobe Analytics found that Americans spent just over $14 billion across online retailers during the events two-day span.
- Amazon's shares fell Thursday alongside other tech stocks.

American shoppers spent big online during Amazon.com's ( AMZN ) two-day Prime Day sales event, according to new estimates.

Amazon, which doesn't detail revenue numbers for its Prime Day sales event , said Thursday that this year's event was the best in the event's 10-year history in terms of sales and items sold. Adobe Analytics said that American shoppers spent about $14.2 billion at online retailers over Tuesday and Wednesday combined.

A record number of customers signed up for Prime in the three weeks leading up to the event, Amazon said. Adobe had earlier this week predicted $14 billion in total online retail spending, up more than 10% from last year's event.

## Back-to-School, Electronics Categories See Gains

This year's event was later in the month than last year's , which may have boosted sales of back-to-school products like backpacks, school supplies, and youth apparel, according to Adobe. Larger purchases like electronics and furniture were also more popular this year, Adobe said.

Sales were split evenly across the two days, Adobe said . Several Amazon retail competitors, including Target ( TGT ), Walmart ( WMT ), and Macy's ( M ) ran their own sales events in the days around Prime Day.

Amazon's shares were recently down more than 2% amid a broader pullback from the big tech stocks that have driven markets this year .

Read the original article on Investopedia .

## How to Watch the 2024 MLB Home Run Derby Tonight: Start Time, TV Channel, Live Stream and More

## This week's MLB All-Star festivities get started at the Home Run Derby tonight. Here's how to watch at home.

The 2024 MLB All-Star Home Run Derby is here. Eight of the biggest names in baseball are assembling at Globe Life Field in Arlington, Texas tonight for one of the summer's most anticipated sporting events. Taking place the night before the annual All-Star Game, this competition is sure to be a prime-time spectacle of strength and endurance that baseball fans won't want to miss.

Watch the Derby on Sling TV

Pete Alonso and Adolis García are back in action after participating in last year’s Home Run Derby. This will be the New York first baseman's fifth straight Derby, including 2019 and 2021 when he won back-to-back. Tonight's Derby winner will receive a whopping $1 million. The runner-up takes home $500K, while every other participant receives $150K. The player who hits the longest home run will win $100K.

Here's everything you need to know about how to watch the MLB Home Run Derby, including the start time, participants and best livestream options.

## How to Watch the 2024 MLB Home Run Derby Without Cable

This year's MLB Home Run Derby will air live on ESPN . If you don't have cable, you can watch the event with a live TV streaming service like Sling TV and FuboTV or stream the Derby on ESPN+.

## Watch the MLB Home Run Derby on Sling TV

The most cost-effective way to watch the 2024 MLB Home Run Derby is Sling TV . Right now Sling TV is offering 50% off your first month of service on any tier, bringing the Orange tier with ESPN down to just $20 and the comprehensive Orange + Blue package to $30.

Sling TV comes with 50 hours of free cloud-based DVR recording space, perfect for recording the Home Run Derby and MLB All-Star Game if you're not home to watch them live. You'll also get access to Fox, FS1, TBS and ABC to catch more MLB games throughout the season. MLB Network is available with Sling TV via the Sports Extra Package.

## MLB Home Run Derby on Sling TV

Sling TV's Orange package offers access to ESPN to watch the 2024 MLB Home Run Derby. Right now, you can get 50% off your first month of Sling TV for just $20.

For Your First Month

## Watch the MLB Home Run Derby for free on FuboTV

Fubo's sports-focused live TV streaming service is another great option for a MLB Home Run Derby livestream. You'll have access to ESPN along with more than 200 other news, entertainment and sports channels, like the MLB Network.

FuboTV comes with 1,000 hours of cloud DVR storage and a seven-day free trial. To watch the Home Run Derby for free, start a seven-day free trial of Fubo and you'll also have access to MLB, NASCAR, Formula 1, MLS, international soccer games and more sporting events happening this summer.

## MLB Home Run Derby on FuboTV

Watch the Home Run Derby without cable on Fubo and stream more of your favorite TV series, movies and sports events on your TV or other devices. You can sign up for a seven-day free trial now to watch the Derby at no cost.

$79.99/Month

7-Day Free Trial

## Watch the MLB Home Run Derby on ESPN+

With ESPN exclusively televising the Home Run Derby, you can also watch the competition on ESPN+. Plans start at $10.99 per month or you can bundle the service with Disney+ and Hulu for $3 more per month. Subscriber get live streaming access to baseball, soccer, golf and more along with on-demand access to original docuseries and ESPN's exclusive sports analysis.

## MLB Home Run Derby on ESPN+

Stream the MLB Home Run Derby on ESPN+ at 8 p.m. ET on July 15. An ESPN+ subscription costs $10.99 per month or $109.99 per year.

## When is the 2024 MLB Home Run Derby?

MLB's annual Home Run Derby will take place on Monday, July 15 at 8 p.m. ET (5 p.m. PT) at Globe Life Field in Arlington, Texas.

## What channel is the 2024 MLB Home Run Derby on?

The Home Run Derby will be broadcast live on ESPN and stream on ESPN+. The 2024 MLB All-Star Game is the following day at the same time and will be broadcast on Fox.

## Who is in the 2024 Home Run Derby?

For this year's Major League Baseball Home Run Derby, the field of eight includes four players from the American League and four from the National League.

- 1B Pete Alonso, New York Mets
- 3B Alec Bohm, Philadelphia Phillies
- OF Adolis García, Texas Rangers
- SS Gunnar Henderson, Baltimore Orioles
- OF Teoscar Hernández, Los Angeles Dodgers
- DH Marcell Ozuna, Atlanta Braves
- 3B Jose Ramirez, Cleveland Guardians
- SS Bobby Witt Jr., Kansas City Royals

Unlike previous years, the first round won't be a head-to-head bracket. All eight hitters will compete against one another in the first round. The four best performers advance to the semifinals and will be seeded based on how many home runs they hit in the first round. If there is a tie, it all comes down to the distance of each hitter's longest home run.

## Is Shohei Ohtani participating in the Home Run Derby?

Shohei Ohtani will not be taking part in the 2024 Home Run Derby. The Dodgers star is is recovering from his injured pitching elbow.

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July 18, 2024

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## Amazon announces record-breaking sales for 2024 Prime Day event

## A behind-the-scenes look at how Amazon prepares for Prime Day, the biggest shopping event of the year

Prime Day 2024 was Amazon’s biggest Prime Day shopping event ever. Here’s how Amazon fulfillment centers and delivery stations prepare.

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## Amazon Prime Day 2024 sales hit record $14.2 billion

Amazon’s Prime Day event clocked record sales this year, as U.S. consumers spent $14.2 billion across July 16 and 17, according to Adobe Analytics . This marks an 11% jump from last year’s $12.7 billion total.

Buyers spent $7.2 billion on day one of the 2024 online event, up 11.2% from a year prior. Adobe Analytics said day two says increased 10.4% year-over-year, hitting $7 billion. While Amazon didn’t release any of its own numbers, the company acknowledged record-breaking sales.

“Prime Day 2024 was a huge success thanks to the millions of Prime members globally who turned to Amazon for fantastic deals, and our much-appreciated employees, delivery partners, and sellers around the world who helped bring the event to life for customers,” Doug Herrington, CEO of Worldwide Amazon Stores, said in a release.

Per Adobe’s report, shopping on phones went up by 18% this year, comprising 49.3% of the total number. Amazon says its Rufus AI assistant helped millions of customers with their shopping .

Analytics site Numerator noted that across two days, the average order value was $57.97. The average order sizes of the past two years’ Prime Day events were $56.64 and $53.14, respectively.

Adobe adds that electronics and furniture drove a lot of sales, with back-to-school shopping also clocking high growth.

“The electronics, apparel, and furniture categories make up nearly half of the e-commerce spend but have seen low single-digit growth in the first half of 2024,” said Vivek Pandya, lead analyst at Adobe Digital Insights.

“It’s clear now that the Prime Day event has been a catalyst across these major categories, with discounts deep enough for consumers to hit the buy button and upgrade items in their homes.”

Adobe said that customers used “buy now pay later (BNPL)” for orders worth $1.08 billion, contributing to 7.6% of sales.

The report predicted that the total consumer spend for July will hit $84.4 billion, up 8.2% from 2023.

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## Recommended

See anne hathaway rock out during taylor swift’s eras tour show in germany.

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Anne Hathaway shook, shook, shook it off at Taylor Swift’s Eras Tour concert in Germany Friday night.

The 41-year-old actress was filmed by a fan moving and grooving to several of the pop star’s hits at her third and final show in Gelsenkirchen. The concertgoer, who shared her videos on X, caught Hathaway rocking out to “…Ready For It?” , “Shake It Off” and “Blank Space.”

The “Devil Wears Prada” star seemed to never tire out from all the fun, as she was seen showing off her moves and throwing her hands excitedly into the air throughout all three videos.

For the show, she wore white shorts and a button-down shirt, accessorizing the look with yellow heart sunglasses.

According to the fan , Hathaway brought one of her sons to watch the Eras Tour with her. Her son reportedly handed out snacks and water bottles to people from the VIP area.

She shares sons Jonathan, 8, and Jack, 4, with her husband, Adam Shulman. The couple chooses to keep them out of the spotlight.

The Academy Award-winning actress has been a longtime supporter of the “Cruel Summer” singer, calling her “a magnificent creature” during a 2015 chat with Glamour magazine .

Of course, Hathway is just one of the several celebrities who have trotted across the globe to catch Swift, 34, on tour. Other stars who have watched the record-breaking show include Selena Gomez, Cara Delevignge, Blake Lively and Ryan Reynolds , Prince William and more.

Her boyfriend, Travis Kelce, has attended several of her shows in Argentina, Australia, Singapore, France, England, Ireland and the Netherlands.

He has also supported her in Germany despite having to report to the Kansas City Chiefs training camp just a day later.

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Join Prime. Watch with Prime. S5 E2 - The Grand Tour: Eurocrash. June 15, 2023. 1 h 46 min. 16+. Jeremy, Richard and James head to Central Europe on a road trip nobody has ever thought of, in cars nobody would ever dream of. This epic 1400-mile journey takes them from Gdańsk in Poland, through Slovakia, Hungary and Slovenia.

Tour de Primes 2024 Welcome to the 15th annual Tour de Primes. 2 is the first prime number...and the only even one. ... The T5K List contains 78 GFN-16 prime numbers, all found in December 2023 and January 2024. Each time a new GFN-16 prime is found, it is added to the T5K List and the 5000th T5K, a GFN-16, drops off the list.

This is a list of articles about prime numbers.A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers.Subsets of the prime numbers may be generated with various formulas for primes.The first 1000 primes are listed below, followed by lists of notable types of prime ...

The Grand Tour is a British motoring television series for Amazon Prime Video, presented by Jeremy Clarkson, Richard Hammond, and James May.The programme focuses on conducting reviews of various models of car, new models and vintage classics, as well as tackling motoring-styled challenges and races, and features the use of studio segments between pre-recorded films.

S3 E1 - Motown Funk. January 17, 2019. 1 h 9 min. TV-14. In the first episode of a brand new season, Jeremy Clarkson, Richard Hammond and James May make a pilgrimage to Detroit to drive three highly tuned muscle cars on the deserted streets of this once-great motor city. Also in this show, Jeremy drives the super-lightweight, super-hardcore ...

December 16, 2020. 1 h 30 min. 16+. The intrepid trio find themselves back on four wheels for their latest adventure. Armed with sports cars, Richard, James and Jeremy think they are in for a cushy road trip as they arrive on the exotic island of Reunion and race on the world's most expensive piece of tarmac.

January 17, 2019. 1 h 9 min. 13+. In the first episode of a brand new season, Jeremy Clarkson, Richard Hammond and James May make a pilgrimage to Detroit to drive three highly tuned muscle cars on the deserted streets of this once-great motor city. Also in this show, Jeremy drives the super-lightweight, super-hardcore, 789 horsepower McLaren Senna.

S4 E1 - The Grand Tour Presents…. Seamen. Watch on supported devices. December 12, 2019. 1 h 32 min. 16+. In the first of a series of feature length Specials, Clarkson, Hammond and May take a one time only break from cars and set out on an epic journey across Cambodia and Vietnam…in boats. This adventure packed voyage sees the hapless trio ...

December 16, 2020. 1 h 30 min. 16+. The intrepid trio find themselves back on four wheels for their latest adventure. Armed with sports cars, Richard, James and Jeremy think they are in for a cushy road trip as they arrive on the exotic island of Reunion and race on the world's most expensive piece of tarmac.

The Grand Tour is a British motoring television series, created by Jeremy Clarkson, Richard Hammond, James May and Andy Wilman, made for Amazon exclusively for its online streaming service Amazon Prime Video, and premiered on 18 November 2016. The programme was conceived in the wake of the departure of Clarkson, Hammond, May and Wilman from the BBC series Top Gear and was originally contracted ...

2 is a Prime Number. Its only factors are 1 and 2. Paste this link in email, text or social media. This prime number calculator will find if a number is prime or composite. If it is composite the calculator will also find all of the factors of the composite number. Factorization or decomposition of composite numbers can be done on the Factoring ...

A prime number is divisible only by itself and 1. There are 25 prime number less than 100. A prime numberis 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.

Here's how you find out if a number is prime. Step 1. Make sure it is a counting number greater than \hspace {0.2em} 1 \hspace {0.2em} 1. And that it isn't a perfect square. Or else, the number cannot be prime. \hspace {0.2em} 97 \hspace {0.2em} 97 passes this test. Step 2.

Means x = 2mn + m + n. that means if x is of form 2mn + m + n then f(x) will give composite numbers otherwise prime no. so reduce the domain of f(x) from N to N − A where A is set of all numbers that can be represented as 2mn + m + n, then we will get prime. Here A can be calculated easily. For example if m = 1, n = 1.

Enter your email address to receive occasional emails about newly added tours and dates, tour changes, and/or special rates and discounts! For booking & Inquiries, or to simply Contact Prime Tours, call 614-766-5553, or email [email protected], or fill out the form on this page!

If you have your heart set on a specific tour, we recommend you make your reservation early. ... through Travelex Insurance Services or contact Travelex Insurance Services at 800-228-9792 and reference location number 35-6094. ... Prime Tours has achieved recognition as a leader in group tourism and is proud to have membership with these elite ...

Question 2: Find all the prime numbers between 20 and 40. Solution: 20 is not a prime number, since it is divisible by 2. 21 is not a prime number, since it is divisible by 3. 22 is not a prime number, since it is divisible by 2. 23 is a prime number since it has no divisors other than 1 and itself. 24 is not a prime number, since it is ...

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At this remove, Pogačar looks destined to ride into Nice in yellow and become the first man since Pantani in 1998 to win the Giro-Tour double, but he - all but - dismissed the prospect of ...

The Levoit Core 300S is a #1 bestseller on Amazon ― and now you can snap up this top-rated air purifier with a 20% discount from a brand we know and trust. When you purchase through links on our ...

Jack Black's comedy rock band Tenacious D has canceled its remaining tour dates after band member Kyle Gass sparked a backlash with an apparent joke about Saturday's assassination attempt on ...

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An American man was arrested ahead of Taylor Swift's Eras Tour show in Gelsenkirchen, Germany, Wednesday night after he allegedly made threats against her and her boyfriend, Travis Kelce. The 34 ...

Amazon, which doesn't detail revenue numbers for its Prime Day sales event, said Thursday that this year's event was the best in the event's 10-year history in terms of sales and items sold.Adobe ...

Prime Video Channels Are on Sale from 99 Cents for Amazon Prime Day. ... How to Watch Every Stage of the 2024 Tour de France. Streaming. How to Watch the 2024 Summer Olympics: Schedule and Live ...

With Amazon's vast selection at some of the lowest prices of the year so far, millions more Prime members shopped the two-day shopping event compared to Prime Day 2023. The opportunity to save big meant a record-breaking number of customers signed up for Prime in the three weeks leading up to Prime Day, with millions of new members worldwide.

Travis Kelce reportedly spent nearly $75,000 on a shopping spree for Taylor Swift ahead of her Eras Tour shows in Milan earlier this month. Getty Images The NFL tight end made stops at Valentino ...

March 21, 2019. 1 h 13 min. TV-14. Jeremy Clarkson, Richard Hammond and James May take an Aston Martin DBS, a Bentley Continental GT and a BMW M850i for an epic drive between the salty Black Sea in Georgia and the fresh water Caspian Sea in Azerbaijan in order to find the best grand touring car for a fish enthusiast.

Amazon's Prime Day event clocked record sales this year, as U.S. consumers spent $14.2 billion across July 16 and 17, according to Adobe Analytics. This marks an 11% jump from last year's $12. ...

Anne Hathaway shook, shook, shook it off at Taylor Swift's Eras Tour concert in Germany Friday night. The 41-year-old actress was filmed by a fan moving and grooving to several of the pop star ...