VMO 2015 Problem 2

For $a,n\in\mathbb{Z}^+$ , consider the following equation: $$a^2x+6ay+36z=n (1)$$ where $x,y,z\in\mathbb{N}$ .

a) Find all $a$ such that for all $n\geq 250$ , $(1)$ always has natural roots $(x,y,z)$ .

b) Given that $a>1$ and $\gcd (a,6)=1$ . Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$ .