I've been working on the following:
Let $m, n$ be two non-zero integers and $p, q$ be two integers. Assume $\gcd(m, n) = 1$. Then show that there exists an integer $N$ such that $m | (N − p)$ and $n | (N − q)$.
I have tried setting $c_1 m + p = c_2 n + q$, but that didn't lead me anywhere. Any tips?
Hint. By Bezout'sidentity, if $m,n$ are two non-zero integers such that $\gcd(m, n) = 1$, then there are integers $x$ and $y$ such that $mx+ny=1$.
Now you are looking for integers $c_1$ and $c_2$ such that $m c_1 + p = n c_2 + q$, that is $m c_1 +n (-c_2)=q-p$.
Can you take it from here?