Different sums that preserve divisibility From my Math GRE book.

If $x$, $y$, and $z$ are integers satisfying $13 | 4x - 5y + 2z$, which one of the following sums must also be divisible by $13$?
(A) x + 13y - z (B) 6x - 10y - z (C) x - y - 2z (D) -7x + 12y + 3z (E) -5x + 3y - 4z

Is there a procedure, given integers $x$, $y$, $z$ and a linear combination $P(x, y, z)$ to write out all of the linear combinations that satisfy $n | P(x, y, z)$?


If $x$, $y$, and $z$ are integers satisfying $13 \mid  4x - 5y + 2z$, which one of the following sums must also be divisible by $13$?
$(A)\ x\! +\! 13y\! -\! z,\ (B)\ 6x\! -\! 10y\! -\! z,\ (C)\ x\! -\! y\! -\! 2z,\ (D)\ \bbox[5px,border:1px solid green]{{\color{#c00}{-7}}x\! +\! 12y\! +\! \color{#c00}3z},\ (E)\ {-}5x\! +\! 3y\!-\! 4z$

Easy way: Let $f = 4x-5y+2z.\,$ If $13\nmid n$ then $13\mid nf\iff 13\mid f,\,$ so it suffices to see if one has the form $nf$. If so the $x$ coef will remain twice the $z$ coef $\!\bmod 13.\,$ Checking only $(D)$ works
$(A)\ \ 1\not\equiv 2(-1),\ \ (B)\ \  6\not\equiv 2(-1),\ \ (C)\ \  1\not\equiv 2(-2),\ \ (D)\ \  \bbox[5px,border:1px solid green]{\color{#c00}{{-}7}\equiv 2(\color{#c00}3)}, \ \ (E)\  {-5}\not\equiv 2(-4)$
$(D)$ is $\,g\equiv 6x\!-\!y\!+\!3z.\,$ $f\,$ has $-5y\,$ vs $-y$ in $g\,$  so $\,5g \equiv 30x -5y +15z\equiv f\pmod{\!13}$ 
