## Greatest Positive Integer $x$

For students of class 9-10 (age 14-16)
Tasnood
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Joined: Tue Jan 06, 2015 1:46 pm

### Greatest Positive Integer $x$

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

samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

### Re: Greatest Positive Integer $x$

Hint
Solution

Mathlomaniac
Posts: 17
Joined: Wed Mar 07, 2018 11:35 pm
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### Re: Greatest Positive Integer $x$

Total times 23 and its multiple appear in 2000! Is 5213
Therefore maximum value of X would be 5213-6=5207

Mathlomaniac
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### Re: Greatest Positive Integer $x$

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

samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

### Re: Greatest Positive Integer $x$

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

soyeb pervez jim
Posts: 21
Joined: Sat Jan 28, 2017 11:06 pm

### Re: Greatest Positive Integer $x$

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 $Legendre's formula$

samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

### Re: Greatest Positive Integer $x$

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

Mathlomaniac
Posts: 17
Joined: Wed Mar 07, 2018 11:35 pm
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### Re: Greatest Positive Integer $x$

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