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.

Guest Sep 27, 2018

#1**+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!.**

Guest Sep 28, 2018

#3**0 **

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

mccartyaiden06 Sep 28, 2018