HMMT 二月 2021 · ALGNT 赛 · 第 1 题

HMMT February 2021 — ALGNT Round — Problem 1

Compute the sum of all positive integers n for which the expression n + 7 √ n − 1 is an integer.

解析

Compute the sum of all positive integers n for which the expression n + 7 √ n − 1 is an integer. Proposed by: Ryan Kim Answer: 89 √ Solution: We know n − 1 must be a positive integer, because the numerator is a positive integer, and the square root of an integer cannot be a non-integer rational. From this, √ n + 7 8 √ = n − 1 + √ n − 1 n − 1 √ is a positive integer, so we n − 1 must be a positive integer that divides 8. There are 4 such positive integers: 1 , 2 , 4 , 8, which give n = 2 , 5 , 17 , 65, so the answer is 89.