HMMT 二月 2024 · ALGNT 赛 · 第 3 题

HMMT February 2024 — ALGNT Round — Problem 3

Compute the sum of all two-digit positive integers x such that for all three-digit (base 10 ) positive integers a b c , if a b c is a multiple of x , then the three-digit (base 10 ) number b c a is also a multiple of x . 3

解析

Compute the sum of all two-digit positive integers x such that for all three-digit (base 10) positive integers a b c , if a b c is a multiple of x , then the three-digit (base 10) number b c a is also a multiple of x . Proposed by: Karthik Venkata Vedula Answer: 64 4 Solution: Note that abc 0 − bca = a (10 − 1) must also be a multiple of x . Choosing a = 1 means that 3 x divides 10 − 1, and this is clearly a necessary and sufficient condition. The only two-digit factors of 3 10 − 1 are 27 and 37, so our answer is 27 + 37 = 64 . 3