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theboyleman32
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This question is a variation on OG PS #116. The most difficult thing is translating what the question is asking.
"m is the product of all the integers from 1 to 40 inclusive" --> This means that m = 40!
"What is the greatest integer p for which 10^p is a factor of m?" --> If we're being asked to maximize the exponent of 10, then the question is really asking: how many factors of 10 are in 40 factorial? Or in other words, how many times does 10 go into 40 factorial?
We definitely don't want to calculate what 40! is - we just need to count the factors of 10.
But we can't simply count 10, 20, 30, and 40. We also need to consider that 2 x 5 = 10, 15 x 4 = 60, etc. What we really need to do is count the 5's, because every factor of 5 will have a corresponding factor of 2. So, just add them up:
5, 10, 15, 20, 25, 30, 35, 40
It looks like 8, but remember - 25 has two factors of 5 in it, so we really have 9 factors of 5, and thus 9 factors of 10 in 40!.
The answer is D.
