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
2
268
3
avatar

What is the total number of Perfect Squares in 100!(100 factorial), that are divisors of that number? Any help would be great. Thank you.

 Sep 27, 2018
 #1
avatar
+1

This is very similar to the one answered some 20 questions below this one, except for its sheer size!
The Perfect Squares are calculated as follows:
First, we have to factor: 100!=2^97×3^48×5^24×7^16×11^9×13^7×17^5×19^5×23^4×29^3×31^3×37^2×41^2×43^2×47^2×53×59×61×67×71×73×79×83×89×97 (239 prime factors, 25 distinct)

Take the exponent of each factor(the ones that have exponents) and divide by 2 and add 1:
(97/2+1)*(48/2+1)*(24/2+1)*(16/2+1)*(9/2+1)*(7/2+1)*(5/2+1)*(5/2+1)*(4/2+1)*(3/2+1)^2*(2/2+1)^4[Take the integer part only]=
=49 * 25 * 13 * 9 * 5 * 4 * 3^3 * 2^6=4,953,312,000 Perfect Squares that are factors(divisors) of 100!.

 Sep 28, 2018
 #2
avatar+102466 
0

Here are the prime factors

I see guest has done the rest. Good work guest :)

 Sep 28, 2018
 #3
avatar-13 
0

"JUST DO YOUR BEST AND NEVER STOP BELIEVING IN SUCESS",I know it sounds cheesy but it will help you alot...wink

 Sep 28, 2018

9 Online Users