Question 6
For positive integers $a,n$, consider
$$a^2x+6ay+36z=n,$$
where $x,y,z$ are natural numbers.
a) Find all $a$ such that for every $n\ge250$ the equation has a solution in natural numbers $(x,y,z)$.
b) Suppose $a\gt 1$ and $\gcd(a,6)=1$. Find, in terms of $a$, the largest $n$ for which the equation has no solution $(x,y,z)$.
$$a^2x+6ay+36z=n,$$
where $x,y,z$ are natural numbers.
a) Find all $a$ such that for every $n\ge250$ the equation has a solution in natural numbers $(x,y,z)$.
b) Suppose $a\gt 1$ and $\gcd(a,6)=1$. Find, in terms of $a$, the largest $n$ for which the equation has no solution $(x,y,z)$.