# What is a prime number? Definition, examples.

## Prime numbers definition

• Positive integers larger than 1 that are only dividing by 1 and themselves are called prime numbers.
• Positive integers that have at least one positive divisor (factor) other than 1 and the number itself are called composite numbers.
A prime number is also any positive integer larger than 1 that is not a composite number.

## Examples of prime numbers. Examples of numbers that are not prime.

• 1 is not considered prime, so the first prime number is 2 (prime numbers list starts by 2);
• 2 is divisible only by 2 and 1, so 2 is a prime number;
• 3 is divisible only by 3 and 1, so 3 is a prime number;
• 4 is divisible by 4, 2 and 1, so 4 is NOT a prime number;
• 5 is divisible only by 5 and 1, so 5 is a prime number;
• 7 is divisible only by 7 and 1, so 7 is a prime number;
• 11 is divisible only by 11 and 1, so 11 is a prime number;
• 12 is divisible by 12, 6, 4, 3, 2 and 1, so 12 is NOT a prime number;
• 13 is divisible only by 13 and 1, so 13 is a prime number;

## All the prime numbers, up to 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.

## Why is 1 not a prime number?

• The fundamental theorem of arithmetic says that every integer larger than 1 can be written as a product of one or more prime numbers in a way that is unique, except for the order of the prime factors.
• Prime numbers are thus the basic building blocks of all numbers.
• If 1 were admitted as a prime, number 15 for example could be factored as 3 × 5 and 1 × 3 × 5; these two representations would be considered different prime factorizations of 15 (prime factorization into prime factors), so the statement of the theorem would have to be modified.