Greatest Positive Integer $x$

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Tasnood
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Greatest Positive Integer $x$

Unread post by Tasnood » Sat Mar 03, 2018 11:30 pm

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

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samiul_samin
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Re: Greatest Positive Integer $x$

Unread post by samiul_samin » Sun Mar 04, 2018 12:04 am

Hint
Use floor function
Answer
$\fbox {83}$
Solution
$23$ is a prime number.

So,we will get $23$ as a divisor of $2000!$

$\lfloor {\dfrac {2000}{23}}\rfloor +\lfloor{\dfrac {2000}{23^2}}\rfloor=89$ times.

So,greatest positive integer $X=89-6=83$
.
So,the answer is $\fbox {83}$

Mathlomaniac
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Re: Greatest Positive Integer $x$

Unread post by Mathlomaniac » 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

Mathlomaniac
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Re: Greatest Positive Integer $x$

Unread post by Mathlomaniac » Thu Mar 08, 2018 10:34 pm

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

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samiul_samin
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Re: Greatest Positive Integer $x$

Unread post by samiul_samin » Sun Mar 25, 2018 6:59 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!
Please watch my solution again.My solution is correct(I think).

soyeb pervez jim
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Re: Greatest Positive Integer $x$

Unread post by soyeb pervez jim » 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

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samiul_samin
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Re: Greatest Positive Integer $x$

Unread post by samiul_samin » 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?

Mathlomaniac
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Re: Greatest Positive Integer $x$

Unread post by Mathlomaniac » Sat Apr 07, 2018 12:44 pm

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.

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