HMMT 二月 2005 · GEN1 赛 · 第 10 题

HMMT February 2005 — GEN1 Round — Problem 10

What is the smallest integer x larger than 1 such that x ends in the same three digits as x does? 1

解析

What is the smallest integer x larger than 1 such that x ends in the same three digits as 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 must divide 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. It is 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