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HMMT 十一月 2015 · 冲刺赛 · 第 15 题

HMMT November 2015 — Guts Round — Problem 15

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 9 ] Find the smallest positive integer b such that 1111 (1111 in base b ) is a perfect square. If no suchbb exists, write ”No solution”.
    HMMT November 2015, November 14, 2015 — GUTS ROUND
    Organization Team Team ID#

英文原题

[ 9 ] Find the smallest positive integer b such that 1111 b (1111 in base b ) is a perfect square. If no such
b exists, write ”No solution”.
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HMMT November 2015, November 14, 2015 — GUTS ROUND
Organization Team Team ID#

解析

英文解析

  1. [ 9 ] Find the smallest positive integer b such that 1111 (1111 in base b ) is a perfect square. If no suchbb 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