Can $n!$ end with exactly five zeros?

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24! ends with four zeros. By adding the factor 25, two more zeros will be added (since the number 25 adds two fives to the prime factorization of the factorial). Therefore, 25! ends with six zeros.

## Answer

How many zeros does the number $100!$ end with?

## Hint

It is sufficient to find how many fives are in the prime factorization of the number 100!.

## Solution

We need to find how many pairs of the prime numbers 2 and 5 are in the prime factorization of 100!. In the factorization of 100!, there will be more twos than fives. Therefore, it is sufficient to find the power to which 5 is raised in the factorization of 100!.

Among the numbers from 1 to 100, there are 20 numbers that contribute fives to the factorization, and the numbers 25, 50, 75, and 100 each contribute two fives. Therefore, the desired power is 24.

## Answer

It ends with 24 zeros.