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

Question

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!

EMPOWERgmatRichC · Accepted Answer

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

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

Marcab · Answer

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.

Bunuel · Answer

Check here: everything-about-factorials-on-the-gmat-85592.html

AKG1593 · Answer

We can count the total number of powers of 2 in 10!
There are 8 of them.So answer D.

Posted from my mobile device

Abhishek009 · Answer

\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

ujjwal80 · Answer

The answer must be D, i.e. 2 will be having a total power of 8 in 10!

JeffTargetTestPrep · Answer

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..

Answer: D

Kinshook · Answer

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

IMO D

CrackverbalGMAT · Answer

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
GMAT SME

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

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

Prep Toolkit

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