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The expression $\frac{1000!}{(500!)^2}$ is an integer. What is the largest integer $n$ such that $7^n$ divides this integer?

Apr 20, 2018

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Since 1000! has 164 factors of 7 and 500! has 82 factors of 7,

$$\frac{1000!}{(500!)^2}$$ has $$164 - 82 - 82 = 0$$

factors of 7.

In other words, all the factors of 7 cancel,

so the greatest power of 7 dividing $$\frac{1000!}{(500!)^2}$$is 7^0 = 1,

so $$n = \boxed{0}.$$

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Apr 20, 2018