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HMMT 十一月 2013 · 冲刺赛 · 第 23 题

HMMT November 2013 — Guts Round — Problem 23

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] The number 989 · 1001 · 1007 + 320 can be written as the product of three distinct primes p, q, rwith p < q < r . Find ( p, q, r ).
解析

英文解析

  1. [ 12 ] The number 989 · 1001 · 1007 + 320 can be written as the product of three distinct primes p, q, rwith p < q < r . Find ( p, q, r ).
    3 2
    Answer: (991 , 997 , 1009) Let f ( x ) = x ( x − 12)( x + 6) + 320 = x − 6 x − 72 x + 320, so thatf (1001) = 989 · 1001 · 1007 + 320. But f (4) = 4( − 8)(10) + 320 = 0, so f ( x ) = ( x − 4)( x − 2 x − 80) =2
    ( x − 4)( x − 10)( x + 8).
    Thus f (1001) = 991 · 997 · 1009, as desired.