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

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Intern
Joined: 21 Jul 2012
Posts: 8
If n is the greatest positive integer for which 2^n is a factor of 10!  [#permalink]

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25 Dec 2012, 09:05
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Difficulty:

5% (low)

Question Stats:

86% (00:47) correct 14% (01:08) wrong based on 481 sessions

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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!
VP
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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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25 Dec 2012, 09:18
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In the questions where you are supposed to find out the maximum power of a prime factor which is a factor of n!, keep one only on thing in mind:
Let the prime number you are looking for be x, then:
$$n/x + n/x^2 + n/x^3.........n/x^k$$ where $$x^k$$ <=n.
If you apply this rule here,
$$10/2 +10/2^2 +10/2^3$$ or 5+2+1=8.
Hence maximum power of 2 in $$10!$$ will be 8.
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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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26 Dec 2012, 02:49
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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Joined: 20 Dec 2013
Posts: 231
Location: India
Re: If n is the greatest positive integer for which 2^n is a factor of 10!  [#permalink]

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09 Jan 2014, 09:44
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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14 May 2015, 22:06
2
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Hi All,

Many Test Takers get these types of questions wrong because they move too quickly through the work and don't do enough work on their pads. It's a relatively straight-forward prompt though...

We're essentially asked to find all the "2"s inside 10!

The 'key' to this question is to realize that some values have MORE THAN ONE 2 in them....

10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1)

10 = 5x2 --> one 2
8 = 2x2x2 --> three 2s
6 = 3x2 --> one 2
4 = 2x2 --> two 2s
2 = 1x2 --> one 2

1+3+1+2+1 = eight 2s

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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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24 Mar 2016, 09:23
1
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]

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03 Apr 2018, 07:29
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]

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05 Apr 2018, 16:12
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! &nbs [#permalink] 05 Apr 2018, 16:12
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