The answer to the question, How Many Trailing Zeros are there in 100 Factorial?, can easily be handed off to a scientific calculator or a CAS(Computer Algebra System). But where is the fun in that? Put another way, we are trying to find the number of zeros at the end of 100 factorial.
The key to solving this problem logically without the aid of technology is to observe that the number of trailing zeros in 100! is the same as the number of 2,5 factor pairs! Since 2 times 5 is 10 which accounts for a single trailing zero, we will be done when have counted all 2,5 factor pairs.
First note that 25, 50,75, and 100 each have two factors of 5 for a total of 8 factors of five.
Also 5,10,15, 20,30,35,40,45,55,60,65,70,80,85,90,95. So these 16 numbers account for 16 more factors of five.
Thus there are 24 factors of five in the expansion of 100 factorial and there are more than enough factors of 2 to pair off with the fives.
Finally we have 24 two,five factor pairs which indicates 100! has 24 trailing zeros.
The proof type of reasoning that was used here was not unlike the thinking that goes on in a chess game. Stimulate your brain with math and chess, not drugs!