What Is a Prime Number? Definition, List up to 100 and Worked Examples

A prime number is one of the most fundamental ideas in mathematics, and it appears constantly, from basic factors to advanced number theory and modern encryption. Despite how often prime numbers come up, a surprising number of students are uncertain about the exact definition, particularly around the edges, such as whether 1 is prime, or how to quickly check a larger number.

This guide gives the precise definition, a complete list of primes to 100, methods for checking primality and the common questions that catch people out.

The definition of a prime number

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.

Two conditions must both be true. The number must be greater than 1, which is why 1 is not prime. And it must have exactly two factors, no more and no fewer, which is why 2 is the only even prime and why any number divisible by anything other than 1 and itself is not prime.

Prime vs composite numbers

Prime	Exactly two factors: 1 and itself	2, 3, 5, 7, 11, 13
Composite	More than two factors	4, 6, 8, 9, 10, 12
Neither	Exactly one factor, which is itself	1 only

The number 1 stands alone. It is neither prime nor composite. This is a deliberate convention, not an oversight, and it is explained in more detail below.

The number 2 is the only even prime number. Every other even number is divisible by 2, giving it at least three factors, which makes it composite. All prime numbers greater than 2 are odd, but not all odd numbers are prime. Nine, fifteen and twenty-five are all odd and all composite.

List of prime numbers from 1 to 100

There are 25 prime numbers between 1 and 100:

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

How to check whether a number is prime

The trial division method

To check whether a number n is prime, test whether it is divisible by any prime up to the square root of n. If no prime up to that point divides it evenly, it is prime.

Why stop at the square root? Because if n has a factor larger than its square root, it must also have a corresponding factor smaller than the square root. So if nothing up to the square root divides it, nothing above will either.

Worked example: is 97 prime?

1.      Find the square root of 97. It is approximately 9.8, so check primes up to 9: that means 2, 3, 5 and 7.

2.      97 is odd, so not divisible by 2.

3.      9 plus 7 equals 16, which is not divisible by 3, so 97 is not divisible by 3.

4.      97 does not end in 0 or 5, so not divisible by 5.

5.      97 divided by 7 is 13.8, not a whole number, so not divisible by 7.

6.      No prime up to the square root divides 97, so 97 is prime.

Worked example: is 91 prime?

7.      Square root of 91 is approximately 9.5. Check 2, 3, 5 and 7.

8.      91 is odd, not divisible by 2.

9.      9 plus 1 equals 10, not divisible by 3.

10.   Does not end in 0 or 5, not divisible by 5.

11.   91 divided by 7 equals 13 exactly. 7 multiplied by 13 equals 91.

12.   91 has the factors 1, 7, 13 and 91. It is not prime, it is composite.

91 is a classic trick question because it looks prime at a glance. It is odd, does not end in 0 or 5 and is not divisible by 2 or 3. The 7 catch trips many students.

The Sieve of Eratosthenes

When you need to find all primes up to a given number, trial division on each one separately is slow. A faster method is the Sieve of Eratosthenes, developed over two thousand years ago.

13.   Write out all whole numbers from 2 to your target.

14.   Start with 2. Circle it as prime, then cross out every multiple of 2: 4, 6, 8, 10 and so on.

15.   Move to the next uncrossed number, which is 3. Circle it as prime, then cross out every multiple of 3 that is not already crossed.

16.   Move to 5, the next uncrossed number. Circle it, cross out its multiples.

17.   Continue until you have worked through all numbers up to the square root of your target.

18.   Every remaining uncircled, uncrossed number is prime.

Working through to 100 requires only checking multiples of 2, 3, 5 and 7, since the square root of 100 is 10 and the only primes up to 10 are 2, 3, 5 and 7.

Why is 1 not a prime number?

This is one of the most common questions in this topic, and the answer matters both for definitions and for how we use primes in practice.

The definition requires exactly two factors. 1 has only one factor, itself. So it fails the definition.

A deeper reason comes from the fundamental theorem of arithmetic, which states that every integer greater than 1 can be written as a unique product of prime factors. For example, 12 equals 2 multiplied by 2 multiplied by 3, and there is only one way to write it that way. If 1 were counted as prime, then 12 could also equal 1 multiplied by 2 multiplied by 2 multiplied by 3, or 1 multiplied by 1 multiplied by 2 multiplied by 2 multiplied by 3, and uniqueness would break. Excluding 1 from the primes preserves the entire structure of number theory.

Why is 2 the only even prime?

Any even number has 2 as a factor. So any even number greater than 2 has at least three factors: 1, 2 and itself. That gives it more than two factors, making it composite. 2 is the only even number with exactly two factors, so it is the only even prime.

Quick divisibility rules that help with prime testing

2	Even number, ends in 0, 2, 4, 6 or 8
3	Sum of digits is divisible by 3
4	Last two digits form a number divisible by 4
5	Ends in 0 or 5
6	Divisible by both 2 and 3
7	No quick shortcut; use division
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
10	Ends in 0

Common exam questions on prime numbers

Find the prime factorisation of 360

Divide repeatedly by the smallest prime that goes in, starting with 2. 360 divided by 2 is 180. 180 divided by 2 is 90. 90 divided by 2 is 45. 45 divided by 3 is 15. 15 divided by 3 is 5. 5 is already prime. So 360 equals 2 to the power 3 multiplied by 3 squared multiplied by 5, written as 2 cubed times 3 squared times 5.

Write 2024 as a product of its prime factors

2024 divided by 2 is 1012. 1012 divided by 2 is 506. 506 divided by 2 is 253. 253 divided by 11 is 23. 23 is prime. So 2024 equals 2 cubed multiplied by 11 multiplied by 23.

Practice questions

19.   List all prime numbers between 50 and 70.

20.   Is 143 prime? Show your working.

21.   Write 180 as a product of its prime factors.

22.   What is the largest prime number less than 50?

23.   What is the smallest prime number that is also even?

Answers

24.   53, 59, 61, 67

25.   The square root of 143 is approximately 11.96, so check primes up to 11: 2, 3, 5, 7, 11. 143 divided by 11 equals 13, so 143 equals 11 multiplied by 13. It is not prime.

26.   180 equals 2 squared multiplied by 3 squared multiplied by 5

27.   47

28.   2

Frequently Asked Questions

Q. Is 0 a prime number?

A. No. The definition requires a whole number greater than 1. Zero is not greater than 1, so it is excluded.

Q. How many prime numbers are there?

A. Infinitely many. This was proved by Euclid more than two thousand years ago. The primes become sparser as numbers get larger, but they never stop.

Q. What is the largest known prime number?

A. As of recent records, the largest known prime has tens of millions of digits. It belongs to a class called Mersenne primes, discovered through distributed computing projects. The record changes periodically.

Q. Is every odd number prime?

A. No. Many odd numbers are composite: 9, 15, 21, 25, 27, 33 and so on. All primes greater than 2 are odd, but the reverse is not true.

Q. What is a twin prime?

A. Two prime numbers that differ by exactly 2, such as 3 and 5, 11 and 13, or 17 and 19. Whether there are infinitely many twin primes is one of the most famous unsolved problems in mathematics.

Disclaimer

This article is for general educational guidance. Always follow the notation and methods required by your own exam board or curriculum.