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HMMT 二月 2017 · 冲刺赛 · 第 27 题

HMMT February 2017 — Guts Round — Problem 27

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Find the smallest possible value of x + y where x, y ≥ 1 and x and y are integers that satisfy 2 2 x − 29 y = 1
解析
  1. [ 15 ] Find the smallest possible value of x + y where x, y ≥ 1 and x and y are integers that satisfy 2 2 x − 29 y = 1 Proposed by: Sam Korsky Answer: 11621 √ 11 16 27 70 2 2 Continued fraction convergents to 29 are 5 , , , , and you get 70 − 29 · 13 = − 1 so since 2 3 5 13 √ √ 2 (70 + 13 29) = 9801 + 1820 29 the answer is 9801 + 1820 = 11621