solution: We may begin by noting that $17\cdot 5=85\equiv 2\pmod{83}$. However, we are looking for $n$ such that $17\cdot n\equiv 1\pmod{83}$.
  
  Note that $2\cdot 42=84\equiv 1\pmod{83}$. Therefore, \begin{align*}
  17\cdot 5\cdot 42 &\equiv 2\cdot 42 \\
  &\equiv 1\pmod{83},
  \end{align*}which tells us that $17$ and $5\cdot 42$ are each other's inverses modulo $83$. We can evaluate $5\cdot 42=210$, but this is not in the range $0$ to $82$, so we take its residue $\pmod{83}$, which is $44$.
  
  Therefore, $17^{-1}\equiv \boxed{44}\pmod{83}$.
  
  We may check our answer: $17\cdot 44 = 748 = 9\cdot 83+1 \equiv 1\pmod{83}$, so our answer is correct.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  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 $17\cdot 5=85$.)
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 44
extra_info.question: 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 $17\cdot 5=85$.)
extra_info.split: test