How might I complete this proof?
Let $(m*n)p = k$ such that p, m, n $\in$ Z. Then $m(n*p) = k$ and $n(m*p) = k$ thus n divides k and m divides k since $n*p$ and $m*p$ must also be integers. Therefore, if $m*n$ divides k, then both m divides k and n divides k.
Could I use the Fundamental Theorem of Arithmetic or Euclid's Lemma alternatively?
The proof is as follows:
Let m, n, and k be integers. Suppose that mn divides k. Then k = (mn)p for some integer p. Note that neither m nor n can be equal to 0. Now, k = (pn)m and k = (pm)n, hence m divides k and n divides k.