Does this got to do with palindromes?

$24x+42y=6(4x+7y)$ . Since $x$ and $y$ are both integers, the answer must be a multiple of 6. The only choice is therefore 6. This can happen when $x=2$ and $y=-1$ .

Bézout's Identity states that

$\exists~ x,y \in Z$ and $x,y \ne 0$ then

$ax+by=d$ where $d=\gcd(a,b)$

So for the equation;

$24x + 42y = d$

$d = \gcd(24,42)$ then

$d = 6$

The answer is $\boxed{6}$