## Tutoring: composite numbers prime factorization (decomposing, breaking numbers down to their 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 theorem would have to be modified.

It is important to know about numbers prime factorization in order to calculate the greatest common factor GCF of numbers (also called the greatest common divizor GCD, or highest common factor, HCF) - GCF is needed when reducing (simplifying) fractions to the lowest terms, or the least common multiple LCM - this is needed when adding or subtracting ordinary fractions...

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; the number itself is called an IMPROPER FACTOR (divisor). Some people also consider 1 as an improper factor.

### To wrap it up: numbers that are dividing only by themselves (improper factor) and 1, are called prime numbers. All the others, except for 0 and 1, are composite numbers (they can be factored).

Example: 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.

**Please find all the prime numbers, up to 100, here:** 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.