Permutation

naturewild · 2006-05-31 18:57:54

Find the largest value of k so that 10^k divides evenly into 100!

To be honest, I dont really understand what the quesiton is asking. Help is appreciated!

luca-deltodesco · 2006-05-31 19:23:01

i think it means, so that 10^k will divide into 100, leaving an integer. but i dont know (in which case it would be 2, 100/10^2 = 1)

Last edited by luca-deltodesco (2006-05-31 19:23:25)

naturewild · 2006-05-31 19:23:59

I thought of that too, but calculator is useless since they will give you "overflow" message.

I'm trying to solve it by paper... but no luck so far.

edit : the 100! is actually 100 factorial. The ! wasnt the ending of the question heh.. it kinda mislead people.

Last edited by naturewild (2006-05-31 19:28:22)

luca-deltodesco · 2006-05-31 19:25:09

what do you mean by overflow?

naturewild · 2006-05-31 19:34:49

Calculator can only do up to 1 x 10^99. Any more of that and it will simply overflow.

100! is an overflow

10^100 is an overlow too.

luca-deltodesco · 2006-05-31 19:50:50

naturewild wrote:

edit : the 100! is actually 100 factorial. The ! wasnt the ending of the question heh.. it kinda mislead people.

in which case i have no idea

Jai Ganesh · 2006-05-31 20:10:32

naturewild,
This is a very interesting question.
The answer is k=24.
This is how it is arrived at.
In 100!, 11 zeros are got from 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100.
There are 10 numbers ending with 5 between 1 and 100, each multiplied by an even number would give an additional zero.
That gives 10 more zeros.
25, 50, 75 have (5x5) as their factor, which in effect means, they are made of two 5s. Hence, add 3 more zeros. Although 100 also as (5x5) as its factor, remember, we have already taken the two zeros of 100 into account.
Thus, 100! would end in 24 zeros.
Hence, 100! would be divisible by 10^24.
This can be verified using the Full Precision Calculator here.
It takes about 138 seconds.

naturewild · 2006-05-31 20:16:32

Thank you for explaining that! It makes sense now.

And 24 is the right answer ^^

Thanks once again!

edit : actually I dont really understand this part.

"25, 50, 75 have (5x5) as their factor, which in effect means, they are made of two 5s. Hence, add 3 more zeros. Although 100 also as (5x5) as its factor, remember, we have already taken the two zeros of 100 into account"

Dont you already take 25, 50, 75 into account?

Last edited by naturewild (2006-05-31 20:26:57)

krassi_holmz · 2006-06-02 05:36:03

Here's a more general fomula:
If p is prime the the gratest power of p which divides n! is given by the formula:

.
Here you are seaching for

Last edited by krassi_holmz (2006-06-02 05:37:40)

krassi_holmz · 2006-06-02 05:40:09

100!=93326215443944152681699
2388562667004907159682643816
2146859296389521759999322991
5608941463976156518286253697
920827223758251185210916864 0000000000 0000000000 0000

Maths rocks, isn't it?

Last edited by krassi_holmz (2006-06-02 05:41:24)

krassi_holmz · 2006-06-02 05:43:50

Another related question:
Let

be the i-th prime.
Find the largest value of k so that 10^k divides

PP-this is easier than the original...

Jai Ganesh · 2006-06-05 00:23:20

krassi_holmz,
the largest value of k such that 10^k divides the product of the first 100 primes is k=1.

krassi_holmz · 2006-06-09 02:35:07

Good

krassi_holmz · 2006-06-10 02:58:02

Why???

Jai Ganesh · 2006-06-10 03:09:42

Coz there's only 1 multiple of 2 and 1 of 5. All the other numbers being multiplied to get the product are prime numbers, and are not multiples of 2 or 5.

krassi_holmz · 2006-06-10 06:08:50

Another answer...
Because the number actually is:
4711930799906184953162487834760260422020574773409675520188634839616415335845034221205
2892567055446819724391040977771579918043802842183150387194449439904925790307206359905
38452312528339864352999310398481791730017201031090

As you see, it ends with 1 null.

krassi_holmz · 2006-06-10 06:10:16

Another challenge...
Find the largest value of k so that 10^k divides:

krassi_holmz · 2006-06-10 06:11:50

Help:
It is greater than 100.

Jai Ganesh · 2006-06-10 17:50:16

200+95+90+85+80+150+70+65+60+55+100+45+40+35+30+50+20+15+10+5

k=1295

I took less than a minute, I am not too sure of the answer

krassi_holmz · 2006-06-10 21:16:35

I don't know... I get 1300...

Math Is Fun Forum

#1 2006-05-31 18:57:54

Permutation

#2 2006-05-31 19:23:01

Re: Permutation

#3 2006-05-31 19:23:59

Re: Permutation

#4 2006-05-31 19:25:09

Re: Permutation

#5 2006-05-31 19:34:49

Re: Permutation

#6 2006-05-31 19:50:50

Re: Permutation

#7 2006-05-31 20:10:32

Re: Permutation

#8 2006-05-31 20:16:32

Re: Permutation

#9 2006-06-02 05:36:03

Re: Permutation

#10 2006-06-02 05:40:09

Re: Permutation

#11 2006-06-02 05:43:50

Re: Permutation

#12 2006-06-05 00:23:20

Re: Permutation

#13 2006-06-09 02:35:07

Re: Permutation

#14 2006-06-10 02:58:02

Re: Permutation

#15 2006-06-10 03:09:42

Re: Permutation

#16 2006-06-10 06:08:50

Re: Permutation

#17 2006-06-10 06:10:16

Re: Permutation

#18 2006-06-10 06:11:50

Re: Permutation

#19 2006-06-10 17:50:16

Re: Permutation

#20 2006-06-10 21:16:35

Re: Permutation

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