How Many Zeros Are There in n factorial

How many trailing zeros are there in 100! (! is read as factorial)? This is one of the most common problems in elementary school and middle school math competitions and for those who have memorized the strategy, this can be solved in less than five seconds. There are (100/5) + (100/25) = 24 trailing zeros in 100!. But why does the trick works?

Small Cases

Example 1: How many zeroes are there in 6!?

For those who are new to the factorial notation, when we say 6!, we mean that we multiply 6 and 5 and 4 all the way down to 1. That is

6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720.

So, where did all the zeros come from? Zero came from 5 multiplied by any even number factor. For example, in 6!,  if we multiply 6 and 5, this will give us 30, a number with one trailing zero. Notice that none of the remaining numbers in the multiplication can add another trailing zero. » Read more