HMMT February

Number theory Difficulty 4.6 AIME Find the answer

How many perfect squares divide $2^{3} \cdot 3^{5} \cdot 5^{7} \cdot 7^{9}$ ?

Your answer

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

Solution

The number of such perfect squares is $2 \cdot 3 \cdot 4 \cdot 5$ , since the exponent of each prime can be any nonnegative even number less than the given exponent.

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