Thanks for posting this - this is one of my favorite concepts that the GMAT tests. 60! is an enormous number - Microsoft Excel would round it off to have probably 3-4 times more zeroes than the correct answer - so you should know from the beginning that they're going to be testing something more along the lines of number theory than anything else.
How do you know how many zeroes a number has? Well, every zero is a factor of 10, so the number of zeroes at the end of an integer will be the same as the number of times that that number is divisible by 10. This question is essentially asking "how many factors of 10 are contained within 60!?"
When you're looking at factor questions, it's often a good idea to break down your number in to its prime factors. The GMAT often tests prime factorization as a concept (in some ways because it's a useful business thought process, too - how can you break a somewhat unique problem in to its basic characteristics/elements?). The prime factorization of 10 is 2 * 5. To be divisible by 10, a number must have a unique pair of 2 and 5.
So, for 60!, we need to find the number of pairs of 2 and 5 that we can break 60! into. Because there will be many more 2s than 5s (every second number is a 2, while every fifth number is a 5), we'll be constrained by the number of 5s. We can find them by listing the multiples of 5 within 60!:
5
10
15
20
25
30
35
40
45
50
55
60
So why is the answer 14 and not 12? Remember that we're breaking each of these numbers into their prime components:
5 = 5
10 = 5 * 2
15 = 5 * 3
20 = 5 * 4
25 = 5 * 5
30 = 5 * 6
35 = 5 * 7
40 = 5 * 8
45 = 5 * 9
50 = 5 * 5 * 2
55 = 5 * 11
60 = 5 * 12
Finding the extra 5s contained within 25 and 50, we can determine that there are 14 5s contained within the number 60!, and we'll have well more than 14 2s, so we have 14 pairings of 5*2, or a total of 14 factors of 10. Therefore, 60! will have 14 zeroes.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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