Question about proof for Euclid's lemma using greatest common divisor

Question

I have been asked to prove the following:

Let p be prime and m,n $\in$ N. If p|mn then p|m or p|n.

My book offers up the following proof:

1. Assume p divides mn but not m. We need to show that p|n. Since p is prime and does not divide m, gcd = (p,m) = 1.

2. By Proposition 6.30: gcd (mn,pn) = n gcd (m,p) = n.

3. Then, by Proposition 6.29 (iii), since p divides both mn and pn, p also divides n. We conclude our proof.

I am having trouble with step 2 of this proof. Specifically, why are we multiplying by n?

Any help would be much appreciated.