We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
228
1
avatar+211 

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
 #1
avatar+982 
+3

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

15 Online Users