Greatest power

Number Theory Level 2

Find the greatest value of x x such that 2 3 6 + x 23^{6+x} divides 2000 ! 2000!


The answer is 83.

Eddie The Head by Eddie The Head , 27, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Stuti Mohan
Apr 30, 2014

There are 86 multiples of 23 upto 2000. (2000/23=86.9, i.e. only 86 multiples) There are 3 multiples of 23^2= 529 upto 2000, which will add 23 in the product 3 more times. Therefore the total no. of times 23 occurs in 2000! = 86+3 = 89 x = 89 - 6 = 83.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Shaurya Chats
Feb 23, 2014

Using Legendre's theorem on factorization of factorials by prime numbers, we can easily get the number of factors as 89.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...