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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 $n = \boxed{0}.$