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.

Answer: A

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