## Tutoring: prime numbers definition

#### Positive integers that are only dividing by themselves and number 1 are called prime numbers.

#### A composite number is a positive integer that has at least one positive divisor other than 1 and the number itself. A composite number is also any positive integer greater than 1 that is not a prime number.

- 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
- 5 is divisible only by 5 and 1, so 5 is a prime number
- 13 is divisible only by 13 and 1, so 13 is a prime number

#### 1 is not considered prime, so **the first prime number is 2** (prime numbers list is starting with the number 2)

##### 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.

- 4 is divisible by 4, 2 and 1, so 4 is not a prime number, it's a composite number; 4 = 2
^{2} - 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it's a composite number; 6 = 2 * 3
- 8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number; 8 = 2
^{3} - 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number; 9 = 3
^{2}

**Examples of 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