Solution #43: 100!
The number of zeros at the end of a number is equal to the number of times
that number is a multiple of 10. However, 5 and 2 are factors of 10.
The number of zeros will therefore be equal to the smaller of the following two
numbers: number of times 2 appears as a factor and number of times 5 appears
as a factor in the breakdown of the original number into prime factors.
Here, of course, 2 appears as a factor more often than 5. Let us calculate the
number of times 5 appears as a factor: 100! has 20 numbers that are multiples
of 5. Some of them (4 of them) are even multiples of 25: 25, 50, 75, and 100.
In the breakdown of 100! into prime factors, one will find 5 raised to the
power 24 (20 + 4 = 24). There are therefore 24 zeros at the end of 100!.
Source: Berrondom, Marie
Categories: Number sense, Factorial, Reasoning, Favorite