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HMMT 二月 2003 · GEN2 赛 · 第 3 题

HMMT February 2003 — GEN2 Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. How many positive rational numbers less than π have denominator at most 7 when written in lowest terms? (Integers have denominator 1.)
解析
  1. How many positive rational numbers less than π have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) Solution: 54 We can simply list them. The table shows that there are 3+3+6+6+12+6+18 = 54. Denominator Values 1 2 3 1 , , 1 1 1 1 3 5 2 , , 2 2 2 1 2 4 5 7 8 3 , , , , , 3 3 3 3 3 3 1 3 5 7 9 11 4 , , , , , 4 4 4 4 4 4 1 2 3 4 6 7 8 9 11 12 13 14 5 , , , , , , , , , , , 5 5 5 5 5 5 5 5 5 5 5 5 1 5 7 11 13 17 6 , , , , , 6 6 6 6 6 6 1 2 6 8 9 13 15 16 20 7 , , . . . , , , , . . . , , , , . . . , 7 7 7 7 7 7 7 7 7