Proving divisibility

Question

Let $x,y$ and $m$ be integers. Prove if $m | 4x$ + y and $m | 7x+2y$ then $m|x$ and $m|y$

Answer 1

Let $u = 4x + y$ and $v = 7x + 2y$ so that $x = 2u - v$ and $y = 4v - 7u$.

Since $m | u$ and $m | v$ it also happens that $m | 2u - v$ and $m | 4v - 7u$.

That proves that $m | x$ and $m | y$.

Answer 2

$\rm\: mod\ m:\ \begin{pmatrix} 4 & 1 \\\\ 7 & 2 \end{pmatrix} \begin{pmatrix} X \\\\ \rm Y\end{pmatrix}\ \equiv\ \begin{pmatrix} 0 \\\\ 0\end{pmatrix}\ \Rightarrow\ \begin{pmatrix} X \\\\ \rm Y\end{pmatrix}\ \equiv\ \begin{pmatrix} 0 \\\\ 0\end{pmatrix}\ $ since the matrix is invertible ($\rm det = 1$)