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# M11-13

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Math Expert
Joined: 02 Sep 2009
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15 Sep 2014, 23:44
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Difficulty:

95% (hard)

Question Stats:

41% (01:06) correct 59% (01:01) wrong based on 128 sessions

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If $$n$$ is an non-negative integer, is $$10^n+8$$ divisible by 18?

(1) $$n$$ is a prime number.

(2) $$n$$ is even.

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Joined: 02 Sep 2009
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15 Sep 2014, 23:44
1
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Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

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10 Aug 2015, 16:34
1
I think this is a high-quality question and I don't agree with the explanation. should say - On the other hand, if n=0 then 10n+8=1+8=9, so in this case 10n+8 is not divisible by "18".
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17 Aug 2015, 02:31
bfwell wrote:
I think this is a high-quality question and I don't agree with the explanation. should say - On the other hand, if n=0 then 10n+8=1+8=9, so in this case 10n+8 is not divisible by "18".

If it's not divisible by 9, then it's also not divisible by 18.
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18 Aug 2015, 05:01
I think this is a high-quality question and I agree with explanation. I do agree with explanation but there is a mistake in it (a typo..)

On the other hand, if n=0 then 10n+8=1+8=9, so in this case 10n+8 is not divisible by 9

It must be :
On the other hand, if n=0 then 10n+8=1+8=9, so in this case 10n+8 is not divisible by 2
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19 Oct 2015, 09:39
Bunuel wrote:
If $$n$$ is an non-negative integer, is $$10^n+8$$ divisible by 18?

(1) $$n$$ is a prime number.

(2) $$n$$ is even.

Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

How can we consider zero for statement when it is non negative and non positive integer.
we can take only 2,4,6,8.................etc
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Joined: 02 Sep 2009
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19 Oct 2015, 21:01
Mechmeera wrote:
Bunuel wrote:
If $$n$$ is an non-negative integer, is $$10^n+8$$ divisible by 18?

(1) $$n$$ is a prime number.

(2) $$n$$ is even.

Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

How can we consider zero for statement when it is non negative and non positive integer.
we can take only 2,4,6,8.................etc

Non-negative integers are those which are not negative: 0, 1, 2, 3, 4, ...
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27 May 2016, 05:21
Bunuel wrote:
Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

Bunuel:

Since any positive value of $$n$$ would result in a divisible (by 18) number, is the explaination for statement 2 supposed to say any negative even number, not positive?

Thanks!
Math Expert
Joined: 02 Aug 2009
Posts: 7036

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27 May 2016, 05:35
Sallyzodiac wrote:
Bunuel wrote:
Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

Bunuel:

Since any positive value of $$n$$ would result in a divisible (by 18) number, is the explaination for statement 2 supposed to say any negative even number, not positive?

Thanks!

Hi,

It is already given that n is a non-negative integer, so 'n' can take value of either 0 or any positive integer..

Statement II tells us that n is even.....
so n can be 0...... then 10^n + 8 = 1+8=9, which is not div by 18.... ans NO
n can be positive EVEN integer, here ans will always be YES

so II is insuff and the EXPLANATION given is correct....
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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27 May 2016, 05:37
chetan2u wrote:
Sallyzodiac wrote:
Bunuel wrote:
Official Solution:

Notice that $$10^n+8$$ is divisible by 18 for any positive value of $$n$$. In this case $$10^n+8=\text{even}+\text{even}=\text{even}$$ so it's divisible by 2. Also, in this case, the sum of the digits of $$10^n+8$$ is 9 so its divisible by 9. Since $$10^n+8$$ is divisible by both 2 and 9, then it's divisible by $$2*9=18$$ too.

On the other hand, if $$n=0$$ then $$10^n+8=1+8=9$$, so in this case $$10^n+8$$ is not divisible by 9.

(1) $$n$$ is a prime number. Hence, $$n$$ is a positive integer. Sufficient.

(2) $$n$$ is even. $$n$$ can be zero as well as any positive even number. Not sufficient.

Bunuel:

Since any positive value of $$n$$ would result in a divisible (by 18) number, is the explaination for statement 2 supposed to say any negative even number, not positive?

Thanks!

Hi,

It is already given that n is a non-negative integer, so 'n' can take value of either 0 or any positive integer..

Statement II tells us that n is even.....
so n can be 0...... then 10^n + 8 = 1+8=9, which is not div by 18.... ans NO
n can be positive EVEN integer, here ans will always be YES

so II is insuff and the EXPLANATION given is correct....

Gotcha, thanks!
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27 Jun 2016, 01:03
This is how I solved it –
Non negative integer means zero, 1,2,3 …….
Now as (1) says that n is prime, so it cannot be zero. Also (2) says that n is even so it cannot be zero. So forget zero.
Take n = 1,2,3…in each case, we will get 100+8 or 10000+8 or 1000000+3 ……
All of these numbers will be divisible by 18 because each of these sums are ending with 8 which is divisible by 2 AND
Sum of all digits is 9 ..(because there are lot of zeros and 1 and 8 in every sum).
As every number is divisible by 9 and 2 …that means it is divisible by 18.
Hence (1) is sufficient. So strike out answers BCE.
Now take (2). N is even. In this case also, it is divisible by 18 due to above mentioned logic. Hence strike out A.
HOLD on! I just read answer by Bunuel and then googled that ZERO is considered as EVEN. Now, I will never forget it.
So (2) is not sufficient because if we take n=0 then we get answer 9 and 9 is not divisible by 18. Hence, answer is A.
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19 Aug 2016, 09:35
Typo in explanation for st 2, kindly rectify
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21 Aug 2016, 02:20
sanghar wrote:
Typo in explanation for st 2, kindly rectify

Can you please tell what typo are you talking about? Thank you.
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21 Aug 2016, 02:27
Bunuel wrote:
sanghar wrote:
Typo in explanation for st 2, kindly rectify

Can you please tell what typo are you talking about? Thank you.

On the other hand, if n = 0, 10^n + 8 is divisible by 9, but not by 2. Hence, not divisible by 18.

Its a minor discrepancy but makes a difference.

Posted from my mobile device
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09 Jun 2017, 06:12
I think this is a high-quality question and I agree with explanation. Small typo - "On the other hand, if n=0n=0 then 10n+8=1+8=910n+8=1+8=9, so in this case 10n+810n+8 is not divisible by 9". It should be "not divisible by 18"
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19 Jun 2017, 21:13
bfwell wrote:
I think this is a high-quality question and I don't agree with the explanation. should say - On the other hand, if n=0 then 10n+8=1+8=9, so in this case 10n+8 is not divisible by "18".

To rule out a given information, we have to get two answers, so the answer explanation is perfect, as it tells when can we get an answer as a "yes" and a "no" with the information given in B.
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22 Jul 2018, 01:21
Missed the 0 while considering even# and marked D!!!
Re: M11-13 &nbs [#permalink] 22 Jul 2018, 01:21
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# M11-13

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