HMMT 十一月 2023 · GEN 赛 · 第 3 题

HMMT November 2023 — GEN Round — Problem 3

Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11 . 2

解析

Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11 . Proposed by: Ankit Bisain, Eric Shen, Pitchayut Saengrungkongka, Sean Li Answer: 72 Solution: Let an arbitrary such number be abcd . Then, we desire 11 | a + b + c + d and 11 | a − b + c − d , where the latter comes from the well-known divisibility trick for 11. Sums and differences of multiples of 11 must also be multiples of 11, so this is equivalent to desiring 11 | a + c and 11 | b + d . As a ∈ [1 , 9] and b, c, d ∈ [0 , 9] , a + c and b + d must be either 0 or 11 (no larger multiple is achievable). There are 8 choices for such ( a, c ) and 9 choices for such ( b, d ) , so the answer is 8 · 9 = 72 . 2