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  • 10! is not divisible by Option: 1 256Option: 2 512<div class='qna-option

10! is not divisible by 

Option: 1

256


Option: 2

512


Option: 3

1024


Option: 4

None of these


Answers (1)

best_answer

The exponent of Prime P in n!

Where [x] stands for greatest integer value of x\epsilon R.

If m is the index of highest power of a prime p that divides n! then 

m=\left[\frac{n}{p} \right ]+\left[\frac{n}{p^{2}} \right ]+\left[\frac{n}{p^{3}} \right ]+.............

 

Now,

As all the options are powers of 2, lets see the total powers of 2 in 10!

Method 1:

\\Powers \,of \,2 \,in\, 10!\\=\left[\frac{10}{2} \right ]+\left[\frac{10}{2^{2}} \right ]+\left[\frac{10}{2^{3}} \right ]+\left[\frac{10}{2^4}\right]............\\ = \left[\frac{10}{2} \right ]+\left[\frac{10}{4} \right ]+\left[\frac{10}{8} \right ]+\left[\frac{10}{16}\right]............\\ \\= 5 + 2 + 1 + 0 + 0..... \\= 8

Hence, 10! is divisible by 28 (=256) and its factors 

So, not divisible by 1024 

 

Method 2

10 ! =10 \times 9\times 8 \times 7\times 6\times 5\times 4\times 3\times 2\times 1

Numbers which consist of factor of 2 are 10 , 8 , 6 , 4 , 2 

10 has one power of 2 

8 has three powers of 2 

6 has one power of 2 

4 has two powers of 2   

2 has one powers of 2 

    \therefore total no. of 2's in 10! is = 8 

    \therefore 10! is divisible by 2 ^8  (=256) and its factors 

So, not divisible by 1024 

Posted by

Irshad Anwar

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