### Brain Teasers

# 1000!

How many trailing zeros are in the number 1000!?

### Answer

The ! stands for factorial. So, there are exactly 249 trailing zeros in 1000!The only way to get a trailing zero is by multiplying a factor of 1000! which is 2 by a factor of 1000! which is 5. The modified problem is thus: How many such pairs of factors are there?

There are 500 = 1000/2 numbers between 1 and 1000 (inclusive) which are even, so there are at least 500 factors of 2 in 1000!. There are 200 = 1000/5 numbers which are multiples of 5. But in addition there are 40 = 1000/25 numbers which are multiples of 25, each of which gives one factor of five in addition to the one counted in the 200 and the one counted in the 40 above. Finally there is 1 = [1000/625] number which is a multiple of 625, this gives one further factor of 5. Thus there are exactly 200 + 40 + 8 + 1 = 249 factors of 5 in 1000! and since there are more than sufficient factors of 2, there are exactly 249 trailing zeros in 1000!

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