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HMMT 二月 2007 · GEN1 赛 · 第 4 题

HMMT February 2007 — GEN1 Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] Let a and b be integer solutions to 17 a + 6 b = 13. What is the smallest possible positive value for a − b ?
解析

英文解析

  1. [ 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 ) ≡
    − 10 (mod 23). Since -4 is an inverse of -6 modulo 23, then we multiply to get ( a − b ) ≡ 17 (mod 23).
    Therefore, the smallest possible positive value for ( a − b ) is 17. This can be satisfied by a = 5, b = − 12.