if $a/b+c/d=n$ then $|b|=|d|$ I have to prove that if gcd$(a,b)=1$ and gcd$(c,d)=1$ and if $a/b+c/d=n\in\mathbb{Z}$ then $|b|=|d|$. Here is my approach:
$$a/b+c/d=\frac{ad+bc}{bd}=n$$
so $bd\mid (ad+bc)$. Also gcd$(a,b)=1$ and gcd$(c,d)=1$ implies gcd$(ad,bc)=$ gcd$(a,c)$gcd$(b,d)$. From here on I dont know what to do. I need hints. Thank you.

(As the OP seems satisfied, I put Daniels argument here for the sake of answering this question)
$bd\mid ad+bc$ implies in particular $b\mid ad+bc$, hence $b\mid ad$. Since $(a,b)=1$, that implies $b\mid d$. One similarly obtains $d\mid b$, so together: $|b|=|d|$.
