How Many Zeroes

The answer is 249.

To start solving problem, we need some basic truth like: When any number is multiplied by 10 it results to addition of a zero at the end of that number.

Take 5 * 10 = 50 , 50 * 10 = 500, 500 * 10 = 5000, and so on.

So if we want to find how many zeros exists at the end of any number, then our target should be multiplication by 10.

Now, what are the factors that makes up number 10? Obviously 1, 2, 5 and 10 itself.

I don’t think there are any more numbers forming 10 except these.

We won’t care about 1 as it gives the same number when we multiply. Then the numbers we need to consider maybe 2, 5 and 10.

Now, lets see inside natural numbers: 1, 2, 3, (2 * 2)=4, 5, (2 * 3)=6, 7, (2 * 2 * 2)=8, 9, (2 * 5) = 10.

So lets figure out solution to this problem too.

During this we could encounter 5, (5 * 5) = 25, (5 * 5 * 5) = 125, (5 * 5 * 5 * 5) = 625.

Now, no more powers to five is possible until 1000 as (5 ^ 5) = 3125 > 1000.

So, if we can divide 1000 with first, second, third and fourth powers of 5 individually then we can find total number of 5 in the whole 1000!. Isn’t this true?

Then, 100051=200 Even if we find decimal value in some condition we only consider integer value. Like in 29.6 we only consider 29. I suppose there is no need of explanation for this.

Similarly, 100052=40

100053=8

And,100054=1.6

so take 1.

So we found the total numbers of 5 in 1000! = 249.

Hence, we simply got the total number of zeros at the end of 1000! which is 249.