
Number Theory
Let n and a be positive integers and let p be a prime number. The p^a is said to exactly divide n. If p^a exactly divides n and p^a+1 does not exactly divide n. Assume p^a exactly divides m and p^b exactly divides n.
What power of p exactly divides m+n. Prove.

Lemma: If exactly divides n, then no greater power of p divides n, and conversely, if and no greater power of p divides n, then exactly divides n.
The answer to your question is obviously a + b, but why? Can you use the lemma to show how?

yes you can use the lemma.

It is clear that if and , then . Suppose for some positive integer k and both exactly divide m and n, respectively. Then since a is the greatest power of p that divides m, it must be true that , which is impossible by the lemma. Therefore, divides mn exactly.
Edit: never mind, I had the question wrong.

I don't think you have enough information.
Consider the following series of integers m, n:
etc.
In each case it is clear that p exactly divides both m and n, but the power of p that exactly divides m + n is not determined by this fact.