Can anyone check my proofs for these two questions:

1. Let

with

. Prove that if

then

2. Let

, prove (by giving a complete proof or a counter example) that, if

then

My Attempt:

1. Since

we know that

for some integer m. Then if we divide both sides by r we are left with

, so p|q.

Is this a correct proof?

Yessssir.
2.

a-b =mk, for some integer k.

And

. So

, and therefore

implies

.

I don't understand what you did in the last lines. It's simpler:

Tonio
Is this right? I'm not 100% convinced if this implication always holds...