If $23|3x+5y$ for some $x,y\in \mathbb{Z}$, then $23|5x-7y$? Let $x,y$ be some integers such that $23|3x+5y$. Show that $23|5x-7y$ as well.
This was my exam question and I did not solve it. Could anyone give some insight how to do this? Even after the exam I can not solve it...

Here's a systematic approach.
Because $\gcd(23,3)=1$, you can get $23\mid x+ay$ like this:
$$23\mid3x+5y\Longrightarrow23\mid8(3x+5y)-23x=x+40y\Longrightarrow23\mid x-6y$$
Then just multiply by 5 and you get what you wanted: $$23\mid5(x-6y)=5x-30y\Longrightarrow23\mid5x-7y$$
