HMMT 十一月 2017 · 团队赛 · 第 1 题

HMMT November 2017 — Team Round — Problem 1

[ 15 ] A positive integer k is called powerful if there are distinct positive integers p , q , r , s , t such that 2 3 5 7 11 p , q , r , s , t all divide k . Find the smallest powerful integer.

解析

[ 15 ] A positive integer k is called powerful if there are distinct positive integers p , q , r , s , t such that 2 3 5 7 11 p , q , r , s , t all divide k . Find the smallest powerful integer. Proposed by: Yuan Yao Answer: 1024 2 3 5 7 First of all, 1024 is powerful because it can be divided evenly by 16 = 256 , 8 = 512 , 4 = 1024 , 2 = 11 128, and 1 = 1. Now we show that 1024 is the smallest powerful number. Since s 6 = t , at least one of them is at least 2. 11 7 If t ≥ 2 or s ≥ 3, then we need the number to be divisible by at least 2 = 2048 or 3 = 2187, which both exceed 1024, so we must have s = 2 and t = 1. If r = 3, then the number must be divisible by 5 7 3 = 243 and 2 = 128, which means that the number is at least 243 · 128 > 1024, so r ≥ 4, and the 5 number is at least 4 = 1024. Therefore the smallest powerful number is indeed 1024.