HMMT 二月 2023 · ALGNT 赛 · 第 5 题

HMMT February 2023 — ALGNT Round — Problem 5

Suppose E , I , L , V are (not necessarily distinct) nonzero digits in base ten for which • the four-digit number E V I L is divisible by 73, and • the four-digit number V I L E is divisible by 74. Compute the four-digit number L I V E .

解析

Suppose E , I , L , V are (not necessarily distinct) nonzero digits in base ten for which • the four-digit number E V I L is divisible by 73, and • the four-digit number V I L E is divisible by 74. Compute the four-digit number L I V E . Proposed by: Sean Li Answer: 9954 Solution: Let E = 2 k and V I L = n . Then n ≡ − 2000 k (mod 73) and n ≡ − k/ 5 (mod 37), so n ≡ 1650 k (mod 2701). We can now exhaustively list the possible cases for k : • if k = 1, then n ≡ 1650 which is not possible; • if k = 2, then n ≡ 2 · 1650 ≡ 599, which gives E = 4 and n = 599; • if k = 3, then n ≡ 599 + 1650 ≡ 2249 which is not possible; • if k = 4, then n ≡ 2249 + 1650 ≡ 1198 which is not possible. Hence, we must have ( E, V, I, L ) = (4 , 5 , 9 , 9), so L I V E = 9954.