The Number of Zeros in 50! (50 Factorial)

The problem to be solved is as follows: How many zeros are at the end of 50! (50 factorial)? Note: 50! = 50*49*48*47*....*4*3*2*1

Here, I offer two solutions.

First Solution: The multiples of ten which are components of 50! (without reusing any of the multiples) are (5*2), 10, 20, 30, 40, 50, (15*4), (25*6), (35*8), (45*12),

Multiplying these together, we have 1.63296*10¹⁷ = 163,296,000,000,000,000

There are 12 zeros in this number - which is the same number of zeros as in 50!, since no other factors of 50! can be multiplied together to produce multiples of ten. Thus, 50! has 12 zeros.

Second Solution: Each multiple of 5 - taken together with some even number - contributes a multiple of 10. Each multiple of 25, however - taken together with some even number - contributes two multiples of 10. So we can count the number of multiples of 5 in 50!. Then we can add another count of the number of multiples of 25 in 50!. There are 2 multiples of 25 in 50! (25 and 50) and 10 multiples of 5 (5, 10, 15, 20, 25, 30, 35, 40, 45, 50). 2+10 = 12. Thus, there are a total of 12 zeros in 50!