HMMT 十一月 2021 · 冲刺赛 · 第 31 题
HMMT November 2021 — Guts Round — Problem 31
题目详情
- [17] For positive integers n , let f ( n ) be the product of the digits of n . Find the largest positive integer m such that ∞ ∑ f ( n ) b log n c 10 m n =1 is an integer.
解析
- [17] For positive integers n , let f ( n ) be the product of the digits of n . Find the largest positive integer
m such that
∞
∑
f ( n )
b log n c
10
m
n =1
is an integer.
Proposed by: Joseph Heerens
Answer: 2070
Solution: We know that if S is the set of all positive integers with
digits, then∑ ∑ f ( n ) f ( n ) (0 + 1 + 2 + ... + 9) = = = b log ( n ) c ` − 1 ` − 1 10 k k k n ∈ S n ∈ S ` ` ( )− 1 45 45 · . k Thus, we can see that ( ) ∞ ∞ ∞− 1 ∑ ∑ ∑ ∑ f ( n ) f ( n ) 45 45 45 k 2025 = = 45 · = = = 45 + . 45 b log ( n ) c b log ( n ) c 10 10 k k − 45 k − 45 k k 1 − k n =1 n ∈ S ` =1 ` =1It is clear that the largest integer k that will work is when k − 45 = 2025 = ⇒ k = 2070 .