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The The product of any two numbers is equal to the product of their greatest common divisor and least common multiple. - Proof
- Prerequisites:
- Proof that the prime factorization of an integer is unique
- Maximum and minimum
- 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.
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:
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.
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 . Recall that the above expressions are equal to .
## See also |