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HMMT 二月 2011 · ALGCALC 赛 · 第 8 题

HMMT February 2011 — ALGCALC Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Find all integers x such that 2 x + x − 6 is a positive integral power of a prime positive integer.
解析

英文解析

  1. Find all integers x such that 2 x + x − 6 is a positive integral power of a prime positive integer.
    Answer: − 3 , 2 , 5 Let f ( x ) = 2 x + x − 6 = (2 x − 3)( x + 2). Suppose a positive integer a divides 2
    both 2 x − 3 and x + 2. Then a must also divide 2( x + 2) − (2 x − 3) = 7. Hence, a can either be 1 orn n