HMMT 十一月 2021 · GEN 赛 · 第 2 题

HMMT November 2021 — GEN Round — Problem 2

Suppose a and b are positive integers for which 8 a b = 27 a b . Find a + b . 32

解析

Suppose a and b are positive integers for which 8 a b = 27 a b . Find a + b . Proposed by: Sean Li Answer: 117 Solution: We have ( ) a b a − b a − b a b 27 a 27 a 27 a b b a 8 a b = 27 a b ⇐⇒ = ⇐⇒ = ⇐⇒ = . b a a − b a b 8 b 8 b 8 3 3 Since 27 = 3 and 8 = 2 , there are only four possibilities: • a/b = 3 / 2 and a − b = 3, which yields a = 9 and b = 6; • a/b = 27 / 8 and a − b = 1, which yields no solutions; • a/b = 2 / 3 and a − b = − 3, which yields a = 6 and b = 9; • a/b = 8 / 27 and a − b = − 1, which yields no solutions. 2 2 2 2 Therefore a + b must equal 6 + 9 = 117. 32