HMMT 二月 2006 · 冲刺赛 · 第 11 题

HMMT February 2006 — Guts Round — Problem 11

[7] Find all positive integers n > 1 for which is the square of a positive integer. n − 1

解析

Find all positive integers n > 1 for which is the square of a positive integer. n − 1 Answer: 5, 37 2 n +7 n +136 8 n +136 144 144 Solution: Write = n + = n + 8 + = 9 + ( n − 1) + . We seek n − 1 n − 1 n − 1 ( n − 1) 2 to find p and q such that pq = 144 and p + q + 9 = k . The possibilities are seen to be 1 + 144 + 9 = 154, 2 + 72 + 9 = 83, 3 + 48 + 9 = 60, 4 + 36 + 9 = 49, 6 + 24 + 9 = 39, 8 + 18 + 9 = 35, 9 + 16 + 9 = 34, and 12 + 12 + 9 = 33. Of these, { p, q } = { 4 , 36 } is the only solution to both equations. Hence n − 1 = 4 , 36 and n = 5 , 37 .