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# What is the largest value of non negative integer N for which 10N is a

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What is the largest value of non negative integer N for which 10N is a  [#permalink]

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Updated on: 17 Jan 2017, 22:30
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63% (01:00) correct 37% (01:08) wrong based on 196 sessions

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What is the largest value of non negative integer N for which $$10^N$$ is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

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Originally posted by ashikaverma13 on 17 Jan 2017, 11:38.
Last edited by abhimahna on 17 Jan 2017, 22:30, edited 1 time in total.
Edited the question.
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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17 Jan 2017, 13:43
Should the question read $$10^N$$ rather than 10N? That's what you'd need in order for the answers to make sense (otherwise, 10N would be a factor for all of these answer choices.)
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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17 Jan 2017, 18:20
1
ashikaverma13 wrote:
What is the largest value of non negative integer N for which 10N is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

Considering the highlighted part to be 10^n

50! has as many 10's as it have 5's
calculating 5's
50!/5=10
50!/25=2
sum=12

Ans C
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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19 Jan 2017, 04:24
rohit8865 wrote:
ashikaverma13 wrote:
What is the largest value of non negative integer N for which 10N is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

Considering the highlighted part to be 10^n

50! has as many 10's as it have 5's
calculating 5's
50!/5=10
50!/25=2
sum=12

Ans C

Could you elaborate a bit more on the reasoning?

why do we have to find the no of 5s and how exactly did it come to 12?
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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19 Jan 2017, 04:36
2
2
ashikaverma13 wrote:
ashikaverma13 wrote:
What is the largest value of non negative integer N for which 10N is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

Could you elaborate a bit more on the reasoning?

why do we have to find the no of 5s and how exactly did it come to 12?

Hi,
The reasoning for these type of Qs is...

What does ! Or factorial mean ..
It means product of all positive integers till that number..
So 50! Means 1*2*3*4*5....*49*50
Now when we multiply 2 and 5 we get a 10..
So 2s will be more than 5s and that is why number of 10s will be equal to number of 5s...

Now how do we find number of 5s in 50!..
When you divide by 5 you get all multiples of 5 till 50..
50/5=10..
These 10 numbers are 5,10,15,20,25,30,35,.40,45,50..
When we divide by 25 we get numbers which had 5^2 in it..
50/25 =2..
And these are 25 Aand 50..
So you have 12 5s or 10s..
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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22 Jan 2018, 06:33
Is there a faster way than counting all the 5s and the 2s?
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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24 Jan 2018, 13:10
So I think the following method is helpful and short too. Finding non-prime factors in factorials (based on a different thread I read).

50/(5^1)+50/(5^2)
=>10+2
Therefore, there are 12 5's in 50.
We know every alternate number is a multiple of 2.

Given the maximim number of 2's and 5's to create a ten can be 12 of each (10^N = 2^12 * 5^12)
The answer is C - 12.

Extra:
Say we calculate the number of 2's in 50 regardless
50/(2^1)+50/(2^2)+50/(2^3)+50/(2^4)
=>25+12+8+3+1
Therefor there are 49 2's in 50.
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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13 Feb 2018, 09:45
chetan2u wrote:
ashikaverma13 wrote:
ashikaverma13 wrote:
What is the largest value of non negative integer N for which 10N is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

Could you elaborate a bit more on the reasoning?

why do we have to find the no of 5s and how exactly did it come to 12?

Hi,
The reasoning for these type of Qs is...

What does ! Or factorial mean ..
It means product of all positive integers till that number..
So 50! Means 1*2*3*4*5....*49*50
Now when we multiply 2 and 5 we get a 10..
So 2s will be more than 5s and that is why number of 10s will be equal to number of 5s...

Now how do we find number of 5s in 50!..
When you divide by 5 you get all multiples of 5 till 50..
50/5=10..
These 10 numbers are 5,10,15,20,25,30,35,.40,45,50..
When we divide by 25 we get numbers which had 5^2 in it..
50/25 =2..
And these are 25 Aand 50..
So you have 12 5s or 10s..

Hi Chetan,

Can you please explain from here onwards "So 2s will be more than 5s and that is why number of 10s will be equal to number of 5s..."

1) why are 2s more than 5s?
2) why do we need to calculate how many 5s are there?

Thanks
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Re: What is the largest value of non negative integer N for which 10N is a  [#permalink]

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16 Feb 2018, 09:39
ashikaverma13 wrote:
What is the largest value of non negative integer N for which $$10^N$$ is a factor of 50!?

A. 5
B. 6
C. 12
D. 15
E. 20

To determine the largest number of N, we need to determine how many times 10 divides 50!. Since 10 breaks into primes of 5 and 2, and since there are there are fewer 5s in 50! than 2s, we can find the number of 5s and thus be able to determine the number of 5-and-2 pairs.

To determine the number of 5s within 50!, we can use the following shortcut in which we divide 50 by 5, then divide the quotient of 50/5 by 5 and continue this process until we no longer get a nonzero quotient.

50/5 = 10

10/5 = 2

Since 2/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!.

Thus, there are 10 + 2 = 12 factors of 5 and hence 12 factors of 10 within 50!

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Re: What is the largest value of non negative integer N for which 10N is a &nbs [#permalink] 16 Feb 2018, 09:39
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