How many zeros end 100 factorial?

Question

The number 100! = 100 	imes 99 	imes 98 	imes \cdots 	imes 2 	imes 1 is gigantic, far too big to write out by hand. Yet you can find how many trailing zeros it ends in with a short calculation. How many zeros does 100! end with?

Accepted Answer

100! ends in exactly 24 zeros. Trailing zeros come from factors of 10. Each trailing zero is one factor of 10 = 2 	imes 5 in the product. So the number of trailing zeros equals how many complete 2	imes5 pairs you can form. In a factorial, factors of 2 are far more common than factors of 5, so the number of 5s is the bottleneck, just count the 5s. Count the factors of 5 among 1 to 100. - Multiples of 5 (5, 10, 15, \dots, 100) each contribute at least one 5: there are \lfloor 100/5 floor = 20 of them. - Multiples of 25 (25, 50, 75, 100) contribute a second 5 each: there are \lfloor 100/25 \…

How many zeros end 100 factorial?

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