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HMMT 二月 2014 · 冲刺赛 · 第 18 题

HMMT February 2014 — Guts Round — Problem 18

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 11 ] Find the number of ordered quadruples of positive integers ( a, b, c, d ) such that a, b, c, and d areall (not necessarily distinct) factors of 30 and abcd > 900.
解析

英文解析

  1. [ 11 ] Find the number of ordered quadruples of positive integers ( a, b, c, d ) such that a, b, c, and d areall (not necessarily distinct) factors of 30 and abcd > 900.
    ( )
    30 30 30 3043
    Answer: 1940 Since abcd > 900 ⇐⇒ < 900, and there are solutions to a b c d 2
    ( )
    2 2 2 4143
    abcd = 2 3 5 , the answer is (8 − ) = 1940 by symmetry.
    2 2