Prove: If the GCD(ab,c)=1, then GCD(a,c)=1 and GCD(b,c)=1.

Question

I'm taking abstract algebra right now and I don't know where to begin with this question.

Answer 1

Since given that $(ab,c)=1$ and to prove $(a,c)=1$. First let $(a,c)=d$, where $d$ is the $gcd$. We have to show $d=1$. Since $d = (a, c)$, we have $d|a$. Hence, $d|ab$. The fact that $(ab,c)=1$ can be rewritten as $abx+cy=1$, where $x$ and $y$ are some integers. Since also $d|c$ (remember that $d = (a, c)$), we have $d|(abx+cy)$ and thus $d|1$; hence $d=1$. Similarly you can do the next one.