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# If n is the greatest positive integer for which 2^n is a factor of 10!

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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
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curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!

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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
We can count the total number of powers of 2 in 10!
There are 8 of them.So answer D.

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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
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curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!

$$\frac{10}{2}$$ = 5
$$\frac{5}{2}$$ = 2
$$\frac{2}{2}$$ = 1

Now, 5+2+1 = 8

Hence the highest power of 2 that will divide 10! is 8
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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
The answer must be D, i.e. 2 will be having a total power of 8 in 10!
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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Let’s put the even numbers from 2 to 10 in prime factors.

2

4 = 2^2

6 = 2 x 3

8 = 2^3

10 = 2 x 5

We see that there are 8 prime factors of 2 in 10!, so 8 is the value of n.

Alternate Solution:

We know that 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. Let’s find all the 2’s in this product.

The number 10 contributes 1 two.

The number 8 contributes 3 twos.

The number 6 contributes 1 two.

The number 4 contributes 2 twos.

The number 2 contributes 1 two.

Thus, we have a total of 8 twos, and so 8 is the value of n..

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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!

Power of 2 in 10! = 5 + 2 + 1 = 8

IMO D
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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
Top Contributor
Solution:

2 being a prime number, repeatedly divide the value by 2 and discard the remainder

Step 1- => 10 by 2 =5

=> 5 by 2 = 2 (Quotient =2 and remainder to be discarded)

=> 2 by 2 = 1 (Quotient =1 remainder to be discarded)

=> 1 by 2 = 0 (Quotient =0)

Step 2-Now add all these quotients from step 1 => 5+2+1 = 8 (option d)

Note:- You cannot divide by non prime numbers in this way, you may have to group them or count separately for the different factors.
For ex,if you have to count number of 6,count number of 3 s and 2s,observe the value which is smaller and that is your answer.

Devmitra Sen
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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
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Re: If n is the greatest positive integer for which 2^n is a factor of 10! [#permalink]
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