divides and greatest common divisor questions

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.

Re: divides and greatest common divisor questions