PUMaC 2013 · 团队赛 · 第 5 题

PUMaC 2013 — Team Round — Problem 5

A palindrome number is a positive integer that reads the same forward and backward. For example, 1221 and 8 are palindrome numbers whereas 69 and 157 are not. A and B are 4-digit palindrome numbers. C is a 3-digit palindrome number. Given that A − B = C , what is the value of C ?

解析

A palindrome number is a positive integer that reads the same forward and backward. For example, 1221 and 8 are palindrome numbers whereas 69 and 157 are not. A and B are 4-digit palindrome numbers. C is a 3-digit palindrome number. Given that A − B = C , what is the value of C ? SOLUTION: A and B must be multiples of 11, so is C from A − B = C . If A and B have the same unit digit, then C has unit digit 0. This contradicts that C is a 3-digit palindrome number. As the difference of A and B is a 3-digit number, their thousands digits (and hence unit digits) can differ by at most 1. So the unit digit of C can only be 1, and that C = 121. For example, we can have 2002 − 1881 = 121. ANSWER: 121