Factorization (or factorisation - British) is taking a composite number apart into numbers that multiply together to get the original number. These smaller numbers are called factors or divisors. 1 is a factor of all numbers.
Prime factorization is breaking apart a composite numbers into prime numbers that can be multiplied to give the larger number. Note that since 1 is not prime, it is not included in the prime factorization.
For example, 12 can be factorized as 4 × 3. Since 4 is not a prime number, that is not its prime factorization. 12's prime factorization is in fact 3 × 2 × 2.
The numbers which are obtained from the factorization are usually ordered, for example, starting with the smallest number. For example, 72=2^3*3^2. The factorization of every number is unique. This generalizes to:
- Every number has a unique prime factorization
- Every prime factorization corresponds to a unique number
Since finding the numbers to multiply together is very difficult for large numbers, this fact can be used in cryptography.