HMMT February

Number theory Difficulty 4.5 AIME Find the answer

How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares?

Your answer

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

Solution

$\left[\sqrt{\frac{10^{9}}{7^{2}}}\right]-\left[\sqrt{\frac{10^{6}}{7^{2}}}\right]=4517-142=4375$ .

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