### Greatest Positive Integer $x$

Posted:

**Sat Mar 03, 2018 11:30 pm**Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

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Posted: **Sat Mar 03, 2018 11:30 pm**

Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

Posted: **Sun Mar 04, 2018 12:04 am**

Posted: **Thu Mar 08, 2018 2:34 pm**

Total times 23 and its multiple appear in 2000! Is 5213

Therefore maximum value of X would be 5213-6=5207

Therefore maximum value of X would be 5213-6=5207

Posted: **Thu Mar 08, 2018 10:34 pm**

Greatest possible integer value of X can be 88783 because 23 is repeated 88789 times in 2000!

Posted: **Sun Mar 25, 2018 6:59 pm**

Please watch my solution again.My solution is correct(I think).Mathlomaniac wrote: ↑Thu Mar 08, 2018 10:34 pmGreatest possible integer value of X can be 88783 because 23 is repeated 88789 times in 2000!

Posted: **Wed Mar 28, 2018 8:18 pm**

In $2000!$ there are $\lfloor {\frac{2000}{23}}\rfloor=86$ numbers which are divisible by $23$

and $\lfloor{{\frac{2000}{23^2}}}\rfloor=3$ numbers which are divisible by $23^2$

so $23^{89}||2000!$

so the answer would be $83$

it a a part of $Legendre's formula$

more information https://en.wikipedia.org/wiki/Legendre%27s_formula

and $\lfloor{{\frac{2000}{23^2}}}\rfloor=3$ numbers which are divisible by $23^2$

so $23^{89}||2000!$

so the answer would be $83$

it a a part of $Legendre's formula$

more information https://en.wikipedia.org/wiki/Legendre%27s_formula

Posted: **Thu Apr 05, 2018 7:26 pm**

Why are you saying it is repeated 88789 times?I know it is wrong but how did you get it?Can you explain?Mathlomaniac wrote: ↑Thu Mar 08, 2018 10:34 pmGreatest possible integer value of X can be 88783 because 23 is repeated 88789 times in 2000!

Posted: **Sat Apr 07, 2018 12:44 pm**

I did a very bad mistake.samiul_samin wrote: ↑Thu Apr 05, 2018 7:26 pmWhy are you saying it is repeated 88789 times?I know it is wrong but how did you get it?Can you explain?Mathlomaniac wrote: ↑Thu Mar 08, 2018 10:34 pmGreatest possible integer value of X can be 88783 because 23 is repeated 88789 times in 2000!