$\displaystyle m=6k+1$.

$\displaystyle n=3k-1$.

$\displaystyle k$ is fixed odd integer.

$\displaystyle a|m$ and $\displaystyle a|n$.

Find all possible values of $\displaystyle a$.

I know that $\displaystyle m$ is odd and $\displaystyle n$ is even. I know

$\displaystyle m=...,-17,-5,7,19,31,..$

$\displaystyle n=...,-10,-4,2,8,14,20,...$

I can find that $\displaystyle a=(6k+1)/q=(3k-1)/r$ (where q and r are any integers). I can even solve for k=[q(3k-1)-r]/[6r]. I guess I can say that when k is an integer in that format, then a divides m and n (or something along those lines). But that's all I've got.

Frankly, I think the answer is $\displaystyle a=\pm1$, but cannot prove it.

BTW, this is a take home quiz, so don't give answers. A hint of where to go would be nice, though.

Edit: My friend suggests Euclidean Algorithm on the definition of $\displaystyle m$ and $\displaystyle n$ (I don't have time to try it now, so I can't give feedback).