HMMT 二月 2017 · ALGNT 赛 · 第 5 题
HMMT February 2017 — ALGNT Round — Problem 5
题目详情
英文原题
- Kelvin the Frog was bored in math class one day, so he wrote all ordered triples ( a, b, c ) of positiveintegers such that abc = 2310 on a sheet of paper. Find the sum of all the integers he wrote down. Inother words, compute
∑
( a + b + c ) ,
abc =2310
a,b,c ∈ Nwhere N denotes the positive integers. 1
解析
英文解析
- Kelvin the Frog was bored in math class one day, so he wrote all ordered triples ( a, b, c ) of positiveintegers such that abc = 2310 on a sheet of paper. Find the sum of all the integers he wrote down. Inother words, compute
∑
( a + b + c ) ,
abc =2310
a,b,c ∈ Nwhere 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 innerabc =2310
sum can be rewritten as
( )
∑
2310
a · τ ,
a | 2310 a
( )
2310
as for any fixed a , there are τ choices for the integers b, c.
( )a
∑
Now consider the function f ( n ) = a · τ . Therefore, f = n ∗ τ , where n denotes the functionna | ng ( n ) = n and ∗ denotes Dirichlet convolution. As both n and τ are multiplicative, f is also multiplica-ative.
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