HMMT 二月 2006 · 冲刺赛 · 第 11 题
HMMT February 2006 — Guts Round — Problem 11
题目详情
英文原题
- [7] Find all positive integers n > 1 for which is the square of a positive integer.
n − 1
解析
英文解析
- Find all positive integers n > 1 for which is the square of a positive integer.
n − 1
Answer: 5, 37
n +7 n +136 8 n +136 144 1442
Solution: Write = n + = n + 8 + = 9 + ( n − 1) + . We seekn − 1 n − 1 n − 1 ( n − 1)
to find p and q such that pq = 144 and p + q + 9 = k . The possibilities are seen to be 2
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 } isthe only solution to both equations. Hence n − 1 = 4 , 36 and n = 5 , 37 .