HMMT 二月 2014 · 冲刺赛 · 第 18 题
HMMT February 2014 — Guts Round — Problem 18
题目详情
- [ 11 ] Find the number of ordered quadruples of positive integers ( a, b, c, d ) such that a, b, c, and d are all (not necessarily distinct) factors of 30 and abcd > 900.
解析
- [ 11 ] Find the number of ordered quadruples of positive integers ( a, b, c, d ) such that a, b, c, and d are all (not necessarily distinct) factors of 30 and abcd > 900. ( ) 3 4 30 30 30 30 Answer: 1940 Since abcd > 900 ⇐⇒ < 900, and there are solutions to a b c d 2 ( ) 3 4 1 2 2 2 4 abcd = 2 3 5 , the answer is (8 − ) = 1940 by symmetry. 2 2