Here's one way to form the prime factorization of a
composite number: use a prime factorization tree. Find two factors of the composite number and put each factor
at the end of a branch. Continue finding factors for each composite number that
"lives" at the end of a branch. A branch stops once a prime factor has been
used. (Why? A prime factor has no factors other than itself and one, so there's
no need to continue branching.) The prime factors are located at the "leaves" of
the tree. The prime factors located on the "leaves" form the prime factorization
of the composite number you started with.
Let's use a factor tree to factor the number 220.
The prime factorization of 220 is found by pulling the "leaves"
(prime numbers) off the branches. So, the prime factorization of 220 = 2 x 2
x 5 x 11 = .
What if we had first used the factors 22 and 10 instead of the
factors 2 and 110? Would we get the same results? Let's see.
Pulling off the "leaves" or prime numbers, we get 220= 2 x 11 x
2 x 5 = . So it does not matter which factors
we start with! The prime factorization will be the same, or we can say the prime
factorization is "unique" (although the factors may be listed in different orders,
as you can see above.)