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What is the greatest integer m for which the

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New post 28 Nov 2013, 07:24
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What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12
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Re: What is the greatest integer m for which the  [#permalink]

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New post 28 Nov 2013, 23:51
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mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12


Here is a conceptual discussion on this question: http://www.veritasprep.com/blog/2011/06 ... actorials/


As for 50!/10^m, the greatest value of m will be obtained by finding the maximum number of 5s in 50!

The quick process for that (discussed in the link above) is
50/5 = 10
10/5 = 2
Total number of 5s is 12. So maximum value of m will also be 12.
Answer (E)
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Re: What is the greatest integer m for which the  [#permalink]

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New post 29 Nov 2013, 01:12
1
mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12



10 = 5 x 2; 2s are in abundance so all we need is to count the number of 5s in 50! and get to the answer.

50! = 1 x 2 x 3 x ....50

5, 10, 15, 20, 25, 25, 30, 35, 40, 45, 50, 50

25 and 50 come twice because they have 2 fives.

There are 12 fives hence the highest 10 power is 12.

The answer is E.

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Re: What is the greatest integer m for which the  [#permalink]

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New post 29 Nov 2013, 02:51
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mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12


Basically we need to find the number of trailing zeros in 50!.

50/5 + 50/5^2 = 10 + 2 = 12. (check here: everything-about-factorials-on-the-gmat-85592.html).

Answer: E.
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Re: What is the greatest integer m for which the  [#permalink]

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New post 29 Nov 2013, 02:51
Bunuel wrote:
mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12


Basically we need to find the number of trailing zeros in 50!.

50/5 + 50/5^2 = 10 + 2 = 12. (check here: everything-about-factorials-on-the-gmat-85592.html).

Answer: E.


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Hope it helps.
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Re: What is the greatest integer m for which the  [#permalink]

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New post 26 Feb 2014, 14:32
Question can be asked as , if 10^m is a factorial of 50!, what is the greatest integer?

My version of answer is:
We can write 10^m = 2^m . 5^m
We can only get 10 when we multiply with 5, so if we solve this problem for 5, we get answer.
As we know formula n/x + n/x^1 + n/x^3 ...until x^k <= n

=> 50/5 + 50/5^2 => 10 + 2 => 12


*(if you are curious, you can solve for 2 also, 50/2 + 50/ 2^1 + 50/2^3 => 25+12+6 =>43. This number is greater than 12. And also you can make only 12 10's , that's the reason to consider 5 )
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Re: What is the greatest integer m for which the  [#permalink]

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New post 04 Nov 2015, 10:43
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mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12


CONCEPT: Power of any Prime Number in any factorial can be calculated by following understanding

Power of prime x in n! = [n/x] + [n/x^2] + [n/x^3] + [n/x^4] + ... and so on
Where,
[n/x] = No. of Integers that are multiple of x from 1 to n
[n/x^2] = No. of Integers that are multiple of x^2 from 1 to n whose first power has been counted in previous step and second is being counted at this step
[n/x^3] = No. of Integers that are multiple of x^3 from 1 to n whose first two powers have been counted in previous two step and third power is counted at this step
And so on.....

Where [n/x] is greatest Integer value of (n/x) less than or equal to (n/x)
i.e. [100/3] = [33.33] = 33
i.e. [100/9] = [11.11] = 11 etc.


10 = 2*5
But power of 5 in in 50! will always be less than power of 2 because multiple of 5 in expansion of 50! will always be lesser than no. of multiples of 2 in expansion of 50!

Hence, Power of 10 will be equivalent to Power of 5 in 50!

Power of 5 in 50! = [50/5] + [50/5^2] + [50/5^3] + ... = 10 + 2 + 0 + 0 + ... = 12

Answer: Option E
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New post 05 Apr 2018, 08:10
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This is a trailing zero question.

By applying trailing zero concept - https://gmatclub.com/forum/everything-a ... 85592.html we can deduce that the answer must be E.
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Re: What is the greatest integer m for which the  [#permalink]

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New post 07 Apr 2018, 04:38
1
mohnish104 wrote:
What is the greatest integer m for which the number 50!/10^m is an integer?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 12


Notice that m is basically the power of 10. let's analyse the character of 10 with exponents.

10^1=10
10^2=100
10^3=1000
10^4=10000

so, one thing is clear that any exponent of 10 increases the no of zeros. ONE important point to be noted that for each zero, one 5 and one 2 are responsible.
50!/10^m will be an integer. to do so we have to eliminate 10^m. 10^m could be 100 , 1000, 1000000 and so on. we will get just zeros in 10^m. As we don't know the value of m it will depend on the no of zero 50! has. it will tell us how many zero we will need to vanish 10^m. Thus we have to find out the no. of zero in 50!.
there is a short-cut to find out zero:

50/5 + 50/(5)^2 = 10+ 2 = 12 . it means will get 12 zero in total in 50!. So, m will be 12.

Now why did we divide 50 by 5 . we have already learnt that each pair of 5 and 2 creates a zero. In 50! we will have no. of 2 as its factor but 5 is limited. So, we found out the no. of 5 in 50!. ultimately, we got 12 fives. thus , 12 pairs of 5 and 2 yield 12 zero.

Thus, the correct answer is E.
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Re: What is the greatest integer m for which the &nbs [#permalink] 07 Apr 2018, 04:38
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