## Tutoring: composite integers numbers prime factorization (decomposing, breaking numbers down into prime factors)

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. 1 is not considered prime, so **the first prime number is 2**. 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 factorizations of 15 into prime numbers, so the statement of the theorem would have to be modified.

Integer numbers prime factorization is important to know in order to perform computing the greatest common factor GCF of numbers (also called the greatest common divizor GCD, or highest common factor, HCF), or the least common multiple LCM, or fractions reducing to lower terms...

A prime number can't be factored into prime factors, but a number that is not prime (a composite number) can be, as it is shown bellow:

120 = 4 * 30 = 2 * 2 * 2 * 15 = 2 * 2 * 2 * 3 * 5 = 2^{3} * 3 * 5

#### If a number is prime, it can not be factored into prime factors, it is divisible only by 1 and itself, which are called IMPROPER FACTORS (divisors).

### Numbers that are dividing only by themselves and by the number 1, are called prime numbers.

2 is divisible only by 2 and 1, so 2 is a prime number; 13 is divisible only by 13 and 1, so 13 is a prime number; 1 is not considered a prime number, so the prime numbers are starting with the number 2 - the first prime number is 2, not 1.

**Examples of all 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