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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 \(n = \boxed{0}.\)

.
 Apr 20, 2018

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