# Prime factorization

 What number do you want to factorize?

All positive whole numbers greater than 2 can be written as a product of primes. This process is called prime factorization.

Small numbers can easily be factorized by hand. For example, the number 24 can be written as a product of the two factors 2 and 12. Then 12 can be written as a product of 2 and 6, and finally 6 can be written as a product of 2 and 3.

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3

It is not possible to factorize further because 2 and 3 are prime numbers. This means that 2 × 2 × 2 × 3 is the prime factorization of the number 24. When multiple factors have the same value it is common to write them using exponents to reduce the size of the expression.

24 = 2^3 × 3

Note that each number has its own unique prime factorization. In other words, it doesn't matter in which order the factorization is done. When the number 24 was factorized earlier we could just as well have started by splitting it up into the factors 4 and 6, but since 4 = 2 × 2 and 6 = 2 × 3 the prime factorization that we get in the end doesn't change.

24 = 4 × 6 = 2 × 2 × 2 × 3 = 2^3 × 3

Prime factorizations can be used to get the greatest common divisor of two or more numbers by calculating the product of the prime factors that all the numbers have in common. For example, the numbers 42 and 70 have the prime factors 2 and 7 in common.

42 = 2 × 3 × 7
70 = 2 × 5 × 7

This means that the greatest common divisor of 42 and 70 is 14.

GCD(42, 70) = 2 × 7 = 14