HMMT 二月 2003 · GEN2 赛 · 第 3 题
HMMT February 2003 — GEN2 Round — Problem 3
题目详情
英文原题
- How many positive rational numbers less than π have denominator at most 7 whenwritten in lowest terms? (Integers have denominator 1.)
解析
英文解析
- How many positive rational numbers less than π have denominator at most 7 whenwritten 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