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# Perfect squares!

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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
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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
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Here are the prime factors

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

Sep 28, 2018
#3
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"JUST DO YOUR BEST AND NEVER STOP BELIEVING IN SUCESS",I know it sounds cheesy but it will help you alot...

Sep 28, 2018