4. [3] Let $a$ and $b$ be integer solutions to $17 a+6 b=13$. What is the smallest possible positive value for $a-b$ ?
Answer: 17. First group as $17(a-b)+23 b=13$. Taking this equation modulo 23 , we get $-6(a-b) \equiv$ $-10(\bmod 23)$. Since -4 is an inverse of -6 modulo 23 , then we multiply to get $(a-b) \equiv 17(\bmod 23)$. Therefore, the smallest possible positive value for $(a-b)$ is 17 . This can be satisfied by $a=5, b=-12$.