HMMT 十一月 2019 · 冲刺赛 · 第 15 题
HMMT November 2019 — Guts Round — Problem 15
题目详情
- [9] Let a, b, c be positive integers such that a b c
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- = 1 . 77 91 143 What is the smallest possible value of a + b + c ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2019, November 9, 2019 — GUTS ROUND Organization Team Team ID#
解析
- [9] Let a, b, c be positive integers such that a b c
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- = 1 . 77 91 143 What is the smallest possible value of a + b + c ? Proposed by: James Lin Answer: 79 We need 13 a + 11 b + 7 c = 1001, which implies 13( a + b + c − 77) = 2 b + 6 c . Then 2 b + 6 c must be divisible by both 2 and 13, so it is minimized at 26 (e.g. with b = 10 , c = 1). This gives a + b + c = 79.