Bunuel wrote:

If the number 200! is written in the form p × 10^q, where p and q are integers, what is the maximum possible value of q?

(A) 40

(B) 48

(C) 49

(D) 55

(E) 64

MANHATTAN GMAT OFFICIAL SOLUTION:To maximize the value of q, we need to constrain p to NOT be a multiple of 10. In other words, q should be the count of every factor of 10 in 200!, and p would simply be the product of the remaining factors of 200!.

To count factors of 10 in 200!, we could start by counting the multiples of 10 between 1 and 200, inclusive. But this method would undercount the number of 10's that are factors of 200!. For example:

200! has 2 and 5 as factors, which multiply to 10, and thus another factor of 10.

200! has 6 and 15 as factors, which multiply to 90, and thus another factor of 10.

200! has 8 and 125 as factors, which multiply to 1000, and thus three more factors of 10.

Therefore, this problem is really about counting the number of 2 × 5 factor pairs that can be made from the factors of 200.

Let's count the number of 5s found among the factors of 200!:

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We have counted not only the multiples of 5, but also the multiples of 25 and 125, as these higher multiples contribute more than one factor of 5 to the total.

There are eight multiples of 25 in our range (namely, 25, 50, 75, 100, 125, 150, 175, and 200). Each of these are also multiples of 5, so we have already counted one factor of 5 for them, and thus each of the eight multiples of 25 contributes one additional factor of 5. Finally, 125 contributes a total of three 5s to our count — but we already counted one factor of 5 when we counted multiples of 5, and another when we counted multiples of 25, leaving only one additional factor of 5 that we need to count for 125.

Thus, the prime factorization for 200! includes 40 + 8 + 1 = 49 factors of 5.

It is fairly easy to see that there are many more factors of 2 in 200!, as there are 100 even numbers between 1 and 200, not to mention the additional 2's coming from the multiples of 4, 8, 16, 32, etc. that contribute extra factors of 2.

Thus, we can only make 49 of the 2 × 5 factor pairs, and the maximum value of q is 49.

The correct answer is C.
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