Question 177
For positive integers $a$, $n$; consider the following equation:
$$
a^2x + 6a y + 36z = n
$$
where $x$, $y$, $z$ are non-negative natural numbers.
a) Find all integers $a$ such that for all $n \ge 250$, the given equation always has natural roots $(x, y, z)$.
b) Given that $a \gt 1$ and $\gcd(a, 6) = 1$. Find the greatest value of $n$ in terms of $a$ such that the given equation doesn't have natural root $(x, y, z)$.
$$
a^2x + 6a y + 36z = n
$$
where $x$, $y$, $z$ are non-negative natural numbers.
a) Find all integers $a$ such that for all $n \ge 250$, the given equation always has natural roots $(x, y, z)$.
b) Given that $a \gt 1$ and $\gcd(a, 6) = 1$. Find the greatest value of $n$ in terms of $a$ such that the given equation doesn't have natural root $(x, y, z)$.