I am trying to prove a conjecture for fun in my abstract algebra class. The specifics of the whole conjecture are not important, but part of my proof requires the following to be true: If $n\mid m$, $x\equiv0\pmod n$, and $x\equiv0\pmod m$, then $x\equiv0\pmod{\frac{m}n}$.
I do want to note that it is only true that $n\mid m$; it is not necessarily true that it's the greatest common divisor of $m$ and $n$. I tried to prove this directly, and did some messing around with the Chinese Remainder Theorem, but I'm not getting anywhere. Does anyone have any ideas on this one? Thanks for your help!
