<s>[INST] $a^2\equiv 10\pmod b$ and $a^3\equiv 33\pmod b$ 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]
 [/INST] 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 $b=1$.
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