SOLUTION: How many zeroes in behind 127!

Question 834159: How many zeroes in behind 127!

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The number of zeros at the right of a large number is the number of factors that we can factor out of that number.
For example,

meaning that there are multiples of in that factorial.
That accounts for some of the 5's in the prime factorization of .
There are more 5's in the prime factorization of , because some of those multiples of have more than one in their factorization.
Some have one extra because they are multiples of ,
and tells us to count more 5's.
To those extra 5's we have to add the second extra 5 of , which has two extra fives.
The total count of 5's in the prime factorization of is
.
With those 5's in the prime factorization, pairing each with one of the many more 2's, we can make factors each equal to .
Those are all the 10's that we can find as factors of , so there are exactly zeros at the end of .
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