HMMT 二月 2005 · GEN1 赛 · 第 10 题

英文解析

10. What is the smallest integer x larger than 1 such that x ends in the same three digitsas x does?
    Solution: 376
    2 3 3
    The condition is that 1000 | x − x = x ( x − 1). Since 1000 = 2 · 5 , and 2 cannot
    3 3
    divide both x and x − 1, 2 = 8 must divide one of them. Similarly, 5 = 125 mustdivide either x or x − 1. We try successive values of x that are congruent to 0 or 1
    modulo 125 and see which ones have the property that x or x − 1 is divisible by 8. Itis easy to check that 125, 126, 250, 251, and 375 do not work, but the next value, 376,
    does, so this is the answer. 3