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100 Factorial

Puzzle:

How many consecutive zeros are there at the end of 100! (100 factorial) ?

For Solution SCROLL DOWN...

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Solution:

There are 24 zeros on the end of 100!

One solution would be to work this out and count the zeros, well I've done this so you don't have to. The answer is...
9332621544394415268169923885626670049071596826438162146859296389521759999322991560894146397615651828625369792082722375825118520916864000000000000000000000000
Well now you can clearly just count the zeros but actually working out the number is not practical so we need another plan.

The clever bit here is thinking what numbers when multiplied together will end in a zero.
So the product of what numbers when multiplied ends in a zero:
1. When one of the things being multiplied ends in zero itself
2. A number ending in 5 multiplied by an even number
3. 25, 50 and 75 when multiplied by some of the small numbers available eg (4, 2 and 6) generate an extra zero

Below is tabulated the origin of all the zeros:
Number Zeros Number Zeros
100 2 95 1
90 1 85 1
80 1 75 2
70 1 65 1
60 1 55 1
50 2 45 1
40 1 35 1
30 1 25 2
20 1 15 1
10 1 5 1

Total

12

Total

12

So that's it then there are 24 zeros at the end of 100!

91 comments:

pg... said...

100->2
95 ->1
90->1
85->1
80->1
75->2
70->1
65->1
60->1
55->1
50->2
45->1
40->1
35->1
30->1
25->2
20->1
15->1
10->1
5-> 1
========
==> 24

Anonymous said...

24

Anonymous said...

100 TWO zeroes
from 20 to 90 add each one - EIGHT zeroes
unit's digit product 9.8.7...2.1 add to one each and hence - TEN zeroes

TOTAL OF TWENTY ZEROES

Rahul Gupta said...

[100/5]+[100/25]
=20+4
=24

Anonymous said...

[100/5]+[100/25]
=20+4
=24

Rahul Gupta said...

[100/5]+[100/25]
=20+4
=24

Rahul Gupta said...

[100/5]+[100/25]
=20+4
=24

Rahul Gupta said...

[100/5]+[100/25]
=20+4
=24

Anonymous said...

For any number result will be:
N/5 +N/25 + N/125 +N/625 +......
Taking integer part only.

Unknown said...

hey satish .. ive to solve : the number of cosecutive zeroes in 2008 .. so i found ur ans.. but was wonderig where do i end the series.. if u cud reply back wud be great.. thanks

Lalitha Mahesh said...

Suman,
Your solution seems to be correct.. but I have a doubt about this:

Why cant any number ending with 5 (other than 25, 75) lead to an extra 0, for example like 45, 65.. and so on.. I am not sure why this cant happen.

Can some body answer this ?

Shyam Bhawsar said...

The reason is 25,50 and 75 contains the full square of 5 i.e. 25 = 5*5*1, 50=5*5*2, 75=5*5*3.

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Sandhya said...

Good one.

Nikhil Aggarwal said...
This comment has been removed by the author.
Nikhil Aggarwal said...

hey dude there is one digit missing in your answer

the last digits of your answer is :
20916864000000000000000000000000
but the correct one is :
210916864000000000000000000000000

Niranjan said...

5! = 120
10 ! = ------- 00

15 ! = ------- 000
20 ! = ------- 0000

in short after every 5 one zero is added. There are 10 5's and 10 100's from 1 to 100 hence 100 ! has 20 Zero's

Anonymous said...

Niranjan

You need to go to 25!
Surprise 6 zeroes - Not 5

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Find My Candy said...

Instead we can calculate it using, ceil(100/5)+ceil(100/(5^2)+ceil(100/5^3) = 24 :)

Unknown said...

Here's a simple math puzzle that forces you to avoid brute force in solving it and instead look for different ways of thinking about the problem.

I thought about which numbers, when multiplied, will give you zeros. There's two types: numbers that end in zero (10, 20, etc...) and numbers that end in 5 (when they're multiplied by an even number!).

Counting all that, you get 21 zeros. But we're not done yet!

There are three instances where you end up with a number that ends in a five, then a zero (for example, I got 150, 50, and 1350 but this was specific for the numbers I multiplied to give me a better illustration of the zeros).

If you multiply these numbers by even numbers, you get one more zero (50 * 2 = 100, 150 * 4 = 600, etc..)

Therefore, there are 24 zeros at the end of 100!

I wonder if you could cleverly figure out what the digit would be right before the zeros started? It must be an even number because it's multiplied by 2, but I wonder if you could circumvent the process of actually solving 100! to find it. THAT would also be a good puzzle.

Unknown said...

It's so simple you guy's are making it tough..................XD
simply divide the 100/5=20
Now divide the 20/5=4
Now add 20+4=24 Got Answer XD...........
Let me explain...........
Number of trailing zero are possible in only two case either in 2 or 5.
But 2 have less number of repetition or power as compare to 5.
So take 5 and divide the factorial number unless your remainder is less than the divisor.....
If you want to make it more clear then watch Rakesh Yadav Publication Videos on youtube...
We are Indian..............XD