HMMT 十一月 2016 · 冲刺赛 · 第 19 题
HMMT November 2016 — Guts Round — Problem 19
题目详情
英文原题
- [ 11 ] Let S be the set of all positive integers whose prime factorizations only contain powers of the
∑
primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in S ). Compute .1
s ∈ Ss
解析
英文解析
- [ 11 ] Let S be the set of all positive integers whose prime factorizations only contain powers of the
∑
primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in S ). Compute .1
s ∈ Ss
Proposed by: Daniel Qu
2017
Answer:
1008
i j
Since every s can be written as 2 · 2017 for non-negative integers i and j , the given sum can be written
∑ ∑
∞ ∞
1 1
as ( )( ). We can easily find the sum of these geometric series since they both havei ji =0 j =0
2 2017
1 1 2 2017 2017
common ratio of magnitude less than 1, giving us ( ) · ) = · = .
1 1
1 2016 1008
1 − 1 −
2 2017