Question

How many zeroes are there at the end of 100!?

Short answer

24

Step-by-step solution

Step 1

By the definition of factorial:

$100!=100·99·98·97·......·3·2·1$

The prime factorization of 10 is 2.5 , thus e then obtain a zero at the end of the number if there is a pair of 2 and 5 in the prime factorization of 100! .

Note that there are much more numbers that are divisible by 2 (and thus 2 in the prime factorization of the number) than numbers divisible by 5.

Then the number of zeros as the end of the number is the same as the number of 5’s in the prime factorization of 100! .

There are 20 numbers between 1 and 100 that are divisible by 5: 5, 10, 15, 20, 25, 30,......, 100

Thus there are 20 numbers that have a 5 in their prime factorization.

However, there are 4 numbers with two 5’s in their prime factorization: 25, 50, 75, 100

In total, there are then 20+4=24 fives in all prime factorizations and thus there are 24 zeros at the end of the number.