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, 2014

Combinatorics Difficulty 5.1 AIME, harder Prove it United States

Problem:
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 $a b c d > 900$ .

Solution

Solution:
Answer: $1940$

Since $a b c d > 900 \Longleftrightarrow \frac{30}{a} \frac{30}{b} \frac{30}{c} \frac{30}{d} < 900$ , and there are $\binom{4}{2}^{3}$ solutions to $a b c d = 2^{2} 3^{2} 5^{2}$ , the answer is $\frac{1}{2}\left(8^{4} - \binom{4}{2}^{3}\right) = 1940$ by symmetry.

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