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HMMT 十一月 2021 · 冲刺赛 · 第 8 题

HMMT November 2021 — Guts Round — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [7] Let p , q , r be primes such that 2 p + 3 q = 6 r . Find p + q + r .
解析
  1. [7] Let p , q , r be primes such that 2 p + 3 q = 6 r . Find p + q + r . Proposed by: Sheldon Kieren Tan Answer: 7 Solution: First, it is known that 3 q = 6 r − 2 p = 2(3 r − p ) , thus q is even. The only even prime is 2 so q = 2 . Further, 2 p = 6 r − 3 q = 3(2 r − q ) , which means that p is a multiple of 3 and thus p = 3 . This means that 2 · 3 + 3 · 2 = 6 r = ⇒ r = 2 . Therefore, p + q + r = 3 + 2 + 2 = 7 .