$a^2\equiv 10\pmod b$ and $a^3\equiv 33\pmod b$

Question

Let $a,b$ be positive integers such that $a^2\equiv 10\pmod b$ and $a^3\equiv 33\pmod b$. What are all possible values of $b$?

We have that $10a\equiv 33\pmod b$, but how does that determine the value of $b$?

[Source: Russian competition problem]

Answer 1

Hint: Try cubing the first and squaring the second

Answer 2

For integers c, d, n, and k we have $c\equiv d \mod n \Rightarrow c^k\equiv d^k \mod n$. Then, using Mark Bennet's hint, we have $a^6\equiv 1000\mod b$ and $a^6\equiv 1089\mod b \Rightarrow 1000+bn=1089+bm\Rightarrow 89=b(n-m)$. But 89 is a prime number, so $b=1$ or $b=89$. $a=30$ and $b=89$ works. Lots of solutions exist for $b=1$.