HMMT 二月 2003 · 冲刺赛 · 第 29 题

HMMT February 2003 — Guts Round — Problem 29

[10] A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5-digit palindrome that is a multiple of 99?

解析

A palindrome is a positive integer that reads the same backwards as forwards, such as
What is the smallest 5-digit palindrome that is a multiple of 99? Solution: 54945 7 Write the number as XY ZY X . This is the same as 10000 X +1000 Y +100 Z +10 Y + X = 99(101 X +10 Y + Z )+20 Y +2 X + Z . We thus want 20 Y +2 X + Z to be a multiple of 99, with X as small as possible. This expression cannot be larger than 20 · 9+2 · 9+9 = 207, and it is greater than 0 (since X 6 = 0), so for this to be a multiple of 99, it must equal 99 or 198. Consider these two cases. To get 198, we must have Y = 9, which then leaves 2 X + Z = 18. The smallest possible X is 5, and then Z becomes 8 and we have the number 59895. To get 99, we must have Y = 4. Then, 2 X + Z = 19, and, as above, we find the minimal X is 5 and then Z = 9. This gives us the number 54945. This is smaller than the other number, so it is the smallest number satisfying the conditions of the problem.