KVPY SA Exam 2015 Question Paper

Question : 5 of 80

Marks: +1, -0

The largest non-negative integer k such that

24^k

divides 13! is

Solution:

Let

13! = 2^m .3^n .λ

When m is maximum possible value
& n is also maximum possible value
So m =

[13/2]+[13/4]+[13/8]+13/16]+....

= 6 + 3 + 1
= 10

n =[13/3]+[13/9]+[13/27]+...

= 2 + 1
= 3
So,

13! = 2^10.3^3.λ.

= 2.(2^3.3) (2^3.3) (2^3.3)λ.

= 2.(24)^3λ.

∴ k = 3

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