Again 2007!
Number Theory
Level
2
Find the number of the trailing zeroes in the product of the first 2007 positive integers.
The answer is 500.
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1 solution
Consider all the prime factors making up 2007!. Each zero we want in our result requires one 2 and one 5. We can easily recognize that we'll have more than enough 2s to go around, and focus only on the 5s.
One-fifth of the numbers rounded down, or 401 of them, will be divisible by five and thus contribute at least one five each to the product.
One-fifth of those rounded down, or 80 of them, will be divisible by 25 and thus contribute a second five each.
One-fifth of those -- 16 -- contribute a third and in the same vein, 3 contribute a fourth.
401 + 80 + 16 + 3 = 500. How fun that the answer itself has a few trailing zeroes!