PUMaC 2017 · 数论（B 组） · 第 6 题
PUMaC 2017 — Number Theory (Division B) — Problem 6
题目详情
- For any integer n ≥ 2, let b be the least positive integer such that, for any integer N , m n divides N whenever m divides the digit sum of N written in base b , for 2 ≤ m ≤ n . Find the n integer nearest to b /b . 36 25 4 2
解析
- We see that b is 1 + lcm(1 , 2 , . . . , n ) where n is the greatest integer less than n such that, n 1+lcm(1 , 2 ,..., 32) lcm(1 , 2 ,..., 32) ′ ′ ′ for some prime p ≤ n , log n ∈ N . 36 = 32 and 25 = 25, so ≈ = p 1+lcm(1 , 2 ,..., 25) lcm(1 , 2 ,..., 25) 3 · 29 · 31 · 2 = 5394 . Problem written by Zack Stier