Given that $33^{-1} \equiv 77 \pmod{508}$, find $11^{-1} \pmod{508}$ as a residue modulo 508.  (Give an answer between 0 and 507, inclusive.)
I notice that $33^{-1}$ is the same as $3^{-1} \cdot 11^{-1}$ modulo 508, since $33 \equiv 3 \cdot 11 \pmod{508}$. So I can rewrite the given equation as $3^{-1} \cdot 11^{-1} \equiv 77 \pmod{508}$. I want to isolate $11^{-1}$, so I need to multiply both sides by the inverse of $3^{-1}$ modulo 508, which is just 3. So I get $11^{-1} \equiv 3 \cdot 77 \pmod{508}$. I get $11^{-1} \equiv 231 \pmod{508}$. 