HMMT 十一月 2016 · 冲刺赛 · 第 19 题

HMMT November 2016 — Guts Round — Problem 19

[ 11 ] Let S be the set of all positive integers whose prime factorizations only contain powers of the ∑ 1 primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in S ). Compute . s ∈ S s

解析

[ 11 ] Let S be the set of all positive integers whose prime factorizations only contain powers of the ∑ 1 primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in S ). Compute . s ∈ S s 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 have i j i =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