Unique Prime Factorization
The Fundamental Theorem of Arithmetic states that every natural number greater than can be written as a product of prime numbers , and that up to rearrangement of the factors, this product is unique . This is called the prime factorization of the number.
can be written as , or , or , or . But there is only one way to write it as a product where all the factors are primes:
This is the prime factorization of , often written with exponents:
For a prime number such as or , the prime factorization is simply itself. Any composite number (that is, a whole number with more than two factors) has a non-trivial prime factorization.
The prime factorization of a number can be found using a factor tree . Start by finding two factors which, multiplied together, give the number. Keep splitting each branch of the tree into a pair of factors until all the branches terminate in prime numbers.
Here is a factor tree for . We start by noticing that is even, so is a factor. Dividing by , we get , and we proceed from there.
This shows that the prime factorization of is .