Compute $17^{-1}\pmod{83}$. Express your answer as a residue from $0$ to $82$, inclusive.

(You may find it helpful to consider the fact that $17\cdot 5=85$.)
Let's use the fact that $17\cdot 5=85$ to find the multiplicative inverse of 17 mod 83. The multiplicative inverse of 17 mod 83 is the number x such that $17x\equiv 1\pmod{83}$. And from the fact that $17\cdot 5=85$, we know that $17\cdot 5\equiv 2\pmod{83}$. So $x=5\cdot 42=210$ is a solution. We need to find the residue of $210$ when divided by $83$. Right. We get $210=2\cdot 83+44$. So the multiplicative inverse of 17 mod 83 is 44. Right. So $17^{-1}\equiv 44\pmod{83}$.

