Proof that the minimum exponent of each factor in the factorization of two integers make up the factorization of their greatest common divisor.
Proof that the maximum exponent of each factor in the factorization of two integers make up the factorization of their least common multiple.
If , then and , then , which is already equal to .
The same goes when . The conclusion is now true when either or is equal to 1.
We now list the prime factorizations of the numbers:
If a factor or is not in the other factorization, we add them to the other factorization by assigning their exponents a 0. Thus, the number of factors should now be equal, and no distinction will be made between the factorizations:
Let's take an example. Suppose . Their factorizations are:
Now since the factor 3 is not in 16, we'll add that to the factorization of 16 with an exponent of 0:
We don't have anything to change in the factorization of 12.
Using the last two proofs stated in the prerequisites, we have:
Evaluating , we would find out that it is just equal to , since one of them would be the minimum and the other would be the maximum, and it still holds when .