HMMT February

Number theory Difficulty 5.1 AIME, harder Find the answer

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 $abcd>900$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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

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