This is one of my favorite problems from book IV.

65. If m is the product of all integers from 1 to 40, inclusive, what is the greatest integer p for which 10^p is a factor of m?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11







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Any time I encounter a problem involving factors, I try to break all relevant numbers into their prime factorizations as simply as possible. In this instance, 10^p can be rewritten as (2*5)^p. We are looking for p. So how many times does (2*5) go into the product 1*2*3*4.....39*40?

Well, for a product of consecutive integers that large, there are going to be a lot more 2s than 5s. Every other number has at least one 2 as a factor, so your 2s are not your limiting factor. In fact, the number of 5s in your product is going to determine p.

Now that have narrowed our search to the numbers which have 5 as a prime factor, we can list them much more easily: 5, 10, 15, 20, 25, 30, 35, 40. There are 8 numbers in our product that have 5 as a factor. At this point, even very smart GMAT test takers choose B and move on. Beware this mistake!

Remember, we aren't being asked how many numbers have 5 as a factor, we are being asked how many 5s are a factor of the total product. Which means that 25, which factors to 5*5, counts twice. There are, in fact, nine 5s in the product of 1 to 40, inclusive. Since there are more than nine even numbers in the product, you can safely determine that p=9. The answer is C.

TAKE-AWAY: There is no partial credit on the GMAT Quant Section. You don't get points for getting 90% of the way to the correct answer. For this reason, you need to always pay very careful attention to the question that is being asked. Shortcuts can be great tools on the GMAT, but only if you keep your eye on exactly what information you are trying to find.