Seems OK.
This was the question proposed to me. I want to see if I am on the right track. If m|n and p|n and gcd(m,p) = 1 then mp|n.
Suppose m|n and p|n and gcd(m,p) = 1. We want to prove that mp|n. This means that n = ms and n = pt and 1 = mu + pv for s,t,u,v element of Z. By multiplying both sides by n we obtain:
1 = mu + pv
n = nmu +npv
n = ptmu + mspv
n = mp(tu + sv)
Since s,t,u,v are elements of Z. Thus mp|n.