How many trailing zeroes are there in the decimal expansion

How many trailing zeroes are there in the decimal expansion of 300!? Please show work!

Solution

Trailing zeros come from two ways:

1. From multiples of 10. Which are:30 in number from 1 to 300 including 300

2. From products of multiples of 2 and 5.

Note number of multiples of 2 are:150 in number

Number of multiples of 5 are: 300/5=60 in number.

So we need only find number of multiples of 5. Each multiple 5 multiplies to another multiple of 2 and gives one zero.

Number of multiples of 5 are given by:

300/5=60

But there are multiples of 25 which give an extra 5 and hence an extra 0

Number of multiples of 25 are:

300/25=12

There are multiples of 125 which give an extra 5 and hence an extra 0

Number of multiples of 125 are:

floor(300/125)=3

So ,total number of trailing zeros are:

60+12+3=75 trailing zeros