HMMT 十一月 2015 · 冲刺赛 · 第 15 题
HMMT November 2015 — Guts Round — Problem 15
题目详情
- [ 9 ] Find the smallest positive integer b such that 1111 (1111 in base b ) is a perfect square. If no such b b exists, write ”No solution”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2015, November 14, 2015 — GUTS ROUND Organization Team Team ID#
解析
- [ 9 ] Find the smallest positive integer b such that 1111 (1111 in base b ) is a perfect square. If no such b b exists, write ”No solution”. Proposed by: Alexander Katz Answer: 7 3 2 2 2 2 We have 1111 = b + b + b + 1 = ( b + 1)( b + 1). Note that gcd( b + 1 , b + 1) = gcd( b + 1 − ( b + 1)( b − b