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# If 13!/2^x is an integer, which of the following represe

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If 13!/2^x is an integer, which of the following represe [#permalink]  23 May 2017, 08:54
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If $$\frac{13!}{2^x}$$ is an integer, which of the following represents all possible values of x?

A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10
[Reveal] Spoiler: OA

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Re: If 13!/2^x is an integer, which of the following represe [#permalink]  22 Sep 2017, 05:48
We should start by the left constraint. 2^0 is 1 and every number is divisible by 1, thus the lower bound is 0, included. Thus, we already remain with two options A and C. Then, we have to check if 10 is included or not. To do so, it is easy to remember that 2^10 = 1024. Thus, we have to check if 13! is divisible by 1024 or 2^10. My idea is to check if there is a way to compute 1024 using the prime factors of 13!.

13! = 1*2*3*4*5*6*7*8*9*10*11*12*13 Then, let's rewrite every number is its prime factors, i.e 12! = 1*2*3*(2*2)*5*(2*3)*7*(2*2*2)*(3*3)*(2*5)*11*(2*2*3)*13. Then it's easy to notice than this long multiplication can be rewritten as 12!=1*2^10*3^5*5^2*7*11*13.

Thus if we divide 13! by 2^10 we see that 2^10 compares in the number above thus it simplifies and what remains is an integer. Thus, 10 is the upper bound of our interval and answer is A!
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Re: If 13!/2^x is an integer, which of the following represe [#permalink]  22 Sep 2017, 08:01
Carcass wrote:

If $$\frac{13!}{2^x}$$ is an integer, which of the following represents all possible values of x?

A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10

We can write 13! = 13*12*11*10*9*8*7*6*5*4*3*2*1

or 13! = $$13 * (3*2^2) * 11 * (2*5) * 9 * (2^3) * 7 * (2*3) * 5 * (2^2) * 3 * (2) * 1$$

So maximum power of 2 = 2^10 ( adding all powers of 2 we get 2^10, so the value of x has to be ≤ 10 to make the fraction as integer)

since $$2^0$$ =1 and it is divisible by any 13! so

we can consider 0 ≤ x ≤ 10
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Re: If 13!/2^x is an integer, which of the following represe [#permalink]  19 Dec 2017, 06:23
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Expert's post
Carcass wrote:

If $$\frac{13!}{2^x}$$ is an integer, which of the following represents all possible values of x?

A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10

Let’s determine the maximum number of factors of 2 within 13!. It would be very time consuming to list out each multiple of 2 in 13!. Instead, we can use the following shortcut in which we divide 13 by 2, and then divide the quotient of 13/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient.

13/2 = 6 (we can ignore the remainder)

6/2 = 3

3/2 = 1 (we can ignore the remainder)

Since 1/2 does not produce a nonzero quotient, we can stop.

The next step is to add our quotients; that sum represents the number of factors of 2 within 13!.

Thus, there are 6 + 3 + 1 = 10 factors of 2 within 13!.

So, x can be between zero and 10 inclusive.

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Re: If 13!/2^x is an integer, which of the following represe   [#permalink] 19 Dec 2017, 06:23
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