HMMT 二月 2011 · 冲刺赛 · 第 5 题

HMMT February 2011 — Guts Round — Problem 5

[ 5 ] Rachelle picks a positive integer a and writes it next to itself to obtain a new positive integer b . 2 For instance, if a = 17, then b = 1717. To her surprise, she finds that b is a multiple of a . Find the b product of all the possible values of . 2 a

解析

[ 5 ] Rachelle picks a positive integer a and writes it next to itself to obtain a new positive integer b . 2 For instance, if a = 17, then b = 1717. To her surprise, she finds that b is a multiple of a . Find the b product of all the possible values of . 2 a Answer: 77 k k k − 1 Suppose a has k digits. Then b = a (10 + 1). Thus a divides 10 + 1. Since a ≥ 10 , we have k 10 +1 k ≤ 11. But since none of 2, 3, or 5 divide 10 + 1, the only possibilities are 7 and 11. These a values are obtained when a = 143 and a = 1, respectively.