Greatest Positive Integer $x$
Greatest Positive Integer $x$
Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$
 samiul_samin
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Re: Greatest Positive Integer $x$
Hint
Answer
Solution

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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 52136=5207
Therefore maximum value of X would be 52136=5207

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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
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Re: Greatest Positive Integer $x$
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!

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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 $Legendre's formula$
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
 samiul_samin
 Posts: 1007
 Joined: Sat Dec 09, 2017 1:32 pm
Re: Greatest Positive Integer $x$
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!

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Re: Greatest Positive Integer $x$
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!