Puzzle:
How many consecutive zeros are there at the end of 100! (100 factorial) ?
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Solution:
There are 24 zeros on the end of 100!
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...
9332621544394415268169923885626670049071596826438162146859296389521759999322991560894146397615651828625369792082722375825118520916864000000000000000000000000Well 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
Below is tabulated the origin of all the zeros:
So that's it then there are 24 zeros at the end of 100!
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!
16 comments:
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
24
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
[100/5]+[100/25]
=20+4
=24
[100/5]+[100/25]
=20+4
=24
[100/5]+[100/25]
=20+4
=24
[100/5]+[100/25]
=20+4
=24
[100/5]+[100/25]
=20+4
=24
For any number result will be:
N/5 +N/25 + N/125 +N/625 +......
Taking integer part only.
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
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 ?
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.
Hi everyone
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Good one.
hey dude there is one digit missing in your answer
the last digits of your answer is :
20916864000000000000000000000000
but the correct one is :
210916864000000000000000000000000
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