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# Factors in Factorials 1

For how many positive integer values of $n$ is $5^n$ a factor of $50!$?

(A) 11      (B) 12      (C) 13      (D) 14      (E) 15

## Translation

Obviously, $5^1$ and $5^2$ are factors of $50!$. We need to know the largest  $n$ such that $5^n$ is a factor of $50!$. It should be clear that we can’t just start dividing $50!$ by larger and larger powers of 5 until we get a non-zero remainder. Factorials get REALLY big REALLY fast! For example,

$10! = 3,629,800$

$50!$ has more than 60 digits!

## Good Enough

This problem isn’t going down to a frontal assault, so we need to find a back door. Another way to think about the question would be “how many fives are in the prime factorization 50 factorial?” Listing the prime factors of the integers between 1 and 50 and checking for numbers with factors of 5 is impractical, but multiples of 5 are easy to pick out, and since there are only 10 multiples of 5 between 1 and 50 listing the multiples of 5 isn’t impractical at all.

 Multiples Number of 5s Total 5 1 1 10 1 2 15 1 3 20 1 4 25 2 6 30 1 7 35 1 8 40 1 9 45 1 10 50 2 12

You have to watch out for factors of 25, but other than that it’s relatively simple and efficient – maybe not 2-minute efficient, but time management is about management, not a set limit for each problem.

## 800

The ten multiples of 5 between 1 and 50 get us 10 factors of 5. However there are other fives hidden in multiples of 25. There are two multiples of 25 between 1 and 50 so we have to count two more fives. Because we are only going up to 50!, this isn’t much quicker than the first method. But if we had a larger factorial, we could start counting 125’s, then 625’s, and so on, allowing us to count much more efficiently than we could with a table or list.

Using this solution on this problem is a bit like using an RPG to kill a large mosquito However, this method is the way to go when confronted with some larger flying pest/predator… maybe a pterodactyl? Anyway, what if the test writers asked you to find the number of positive integer values of $n$ such that $5^n$ is a factor of $1000!$? I’ll let you try this one yourself. Watch out for hidden fives! Hopefully, you’ll find 249 5’s.

The correct answer is (B)