Writing direct proofs Let $x$, $y$ be elements of $\mathbb{Z}$. Prove if $17\mid(2x+3y)$ then $17\mid(9x+5y)$.  Can someone give advice as to what method of proof should I use for this implication? Or simply what steps to take?

I would write the equations in $\mathbb{Z}_{17}$, which is a field, because $17$ is prime, so linear algebra applies:
$$ 2x+3y=0 $$
is a linear equation of two variables, and you seek to prove that it implies
$$ 9x+5y=0 $$
which means they're linearly dependent. Two equations are linearly dependent if and only if one is a multiple of the other - and this should be easy to prove.
Edit: Since you asked about proof strategy, I'd like to emphasize that this is not some random trick; the condition $p|x$ is not very nice to work with algebraically, but because $\mathbb{Z}_p$ is a field, the equivalent statement $x\equiv 0\mod p$ (I omitted the $\mod 17$ and the $\equiv$ above to make it look more like familiar algebra) is much simpler and better, because you can multiply, add and create linear spaces over $\mathbb{Z}_p$ that behave (in many ways) like real numbers.
