HMMT 二月 2017 · ALGNT 赛 · 第 5 题

HMMT February 2017 — ALGNT Round — Problem 5

Kelvin the Frog was bored in math class one day, so he wrote all ordered triples ( a, b, c ) of positive integers such that abc = 2310 on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute ∑ ( a + b + c ) , abc =2310 a,b,c ∈ N where N denotes the positive integers. 1

解析

Kelvin the Frog was bored in math class one day, so he wrote all ordered triples ( a, b, c ) of positive integers such that abc = 2310 on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute ∑ ( a + b + c ) , abc =2310 a,b,c ∈ N where N denotes the positive integers. Proposed by: Yang Liu Answer: 49140 ∑ Note that 2310 = 2 · 3 · 5 · 7 · 11 . The given sum clearly equals 3 a by symmetry. The inner abc =2310 sum can be rewritten as ( ) ∑ 2310 a · τ , a a | 2310 ( ) 2310 as for any fixed a , there are τ choices for the integers b, c. a ( ) ∑ n Now consider the function f ( n ) = a · τ . Therefore, f = n ∗ τ , where n denotes the function a | n a g ( n ) = n and ∗ denotes Dirichlet convolution. As both n and τ are multiplicative, f is also multiplica- tive. It is easy to compute that f ( p ) = p + 2 for primes p . Therefore, our final answer is 3(2 + 2)(3 + 2)(5 + 2)(7 + 2)(11 + 2) = 49140 . 1