My professor proposed a question in class awhile ago and told us to prove it for homework. When he proved it in class the day it was due, he gave a very long and unnecessary proof. My proof was significantly shorter and when I asked my professor if my proof was correct he told me to go away, so, since I have such a helping professor, I have to know if this proof is correct so that I can study for the exam later on using the right information. Here's the statement:

If , then

Where (,) is the gcd and [,] is the lcm.

Proof:

Suppose there are two integers s and t s.t.

and

This implies that:

Where

Without loss of generality, assume ,

This implies that

We know that since is the lcm by definition, and that

This implies that

Therefore

, where and

and by definition. Since they both are exactly divisible by the same power of p, they must be equal.

Thanks for the help.