Prove that for any number $\displaystyle n$ $\displaystyle (10n+3,5n+2)=1$

The way I approached it was by linear combinations

$\displaystyle d=m(10n+3)+q(5n+2)$

$\displaystyle =10mn+3m+5qn+2q$

$\displaystyle =5n(2m+q)+3m+2q$

At this point I get stuck. I kind of wish there were some way of saying since it was a linear combination of 5,3, and 2 the gcd of the three must be one.