Abstract algebra problems

[trickslapper]

Let a,b,c be integers such that a does not equal 0. Prove that if a|bc, then a|c*gcd(a,b)



gcd(a,b) stands for the greatest common denominator of integers a and b.

I figured this one out i think: gcd(a,b) can be written as am+bn by definition and then c*(am+bn)=acm*bcn, and a*j(where j is some integer) = bc because of our assumption, so then we have: acm*ajn and therefore a|c*gcd(a,b)

does that seem right? the book doesn't give answers for questions that require proofs so i'm not sure either way.. gonna work on the next one now.
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Prove that gcd(ab,c) =1 if and only if (a,c)=1 and (b,c)=1.



there are more problems but they are similar, i'm not sure what to do with the first problem, but im pretty sure that second problem deals with relatively prime numbers. Any ideas?

 

[daon]

gcd(ab,c) = 1 if and only if there are p,q s.t. pab+cq=1. that is, a[pb]+c[q] = 1 and b[ap] + c[q]=1.