HMMT 二月 2011 · 冲刺赛 · 第 5 题

英文解析

5. [ 5 ] Rachelle picks a positive integer a and writes it next to itself to obtain a new positive integer b .
   For instance, if a = 17, then b = 1717. To her surprise, she finds that b is a multiple of a . Find the 2
   product of all the possible values of .ba 2
   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 havek
   10 +1 k
   ≤ 11. But since none of 2, 3, or 5 divide 10 + 1, the only possibilities are 7 and 11. Theseavalues are obtained when a = 143 and a = 1, respectively.