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Intern  Joined: 20 Jun 2011
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If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Question Stats: 53% (01:50) correct 47% (02:05) wrong based on 420 sessions

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If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6

Originally posted by superpus07 on 23 Jul 2012, 10:59.
Last edited by Bunuel on 14 Aug 2012, 23:22, edited 1 time in total.
Edited the question.
Intern  Joined: 21 Sep 2010
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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1
Great question. Made me think for a while.

My answer is D.) "K is a multiple of 9".

We know that M is always divisible by 18 which means that M is always divisible by 9. This implies that the sum of the digits of M (also referred to as k) will ALWAYs be divisible by 9 (refer to the divisibility rules if you want confirmation).

I kept thinking 0 was a loophole until I realized that 0 is a multiple of ALL integers so in the case k=0 (occurs when n=0), k is still a multiple of 9. I also got stuck for a bit on answer choice E.) "K is a multiple of 6" until some plug-n-chug at n=1 disproved this answer.

I'd be very interested in seeing how other people solved this - please post if you used a different route of thinking.
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Club909 wrote:
Great question. Made me think for a while.

My answer is D.) "K is a multiple of 9".

We know that M is always divisible by 18 which means that M is always divisible by 9. This implies that the sum of the digits of M (also referred to as k) will ALWAYs be divisible by 9 (refer to the divisibility rules if you want confirmation).

I kept thinking 0 was a loophole until I realized that 0 is a multiple of ALL integers so in the case k=0 (occurs when n=0), k is still a multiple of 9. I also got stuck for a bit on answer choice E.) "K is a multiple of 6" until some plug-n-chug at n=1 disproved this answer.

I'd be very interested in seeing how other people solved this - please post if you used a different route of thinking.

D says that k has to be a multiple of 9
K = sum of the digits of M

so lets a couple of cases

we know m = 18 n

when n=0 m=0 so k =0 and k is a multiple of 9 , D is true

when n=1,2...6 m= 18, 36,....108 so k = 9 again K is a multiple of 9 , D is again true

when n = -1 or -2 or -6 then m = -18 or -36 or -108 then k = 7 or 3 or 7 ..but now K is not a multiple of 9 ??

so how can D always be true ??

Please note question does not mention that n is a positive integer or M is a positive integer .
if n is a negative integer as shown above then m will be negative and the sum of the digits of M will not always be 9 so please do explain
how D is always true ??
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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stne wrote:
Club909 wrote:
Great question. Made me think for a while.

My answer is D.) "K is a multiple of 9".

We know that M is always divisible by 18 which means that M is always divisible by 9. This implies that the sum of the digits of M (also referred to as k) will ALWAYs be divisible by 9 (refer to the divisibility rules if you want confirmation).

I kept thinking 0 was a loophole until I realized that 0 is a multiple of ALL integers so in the case k=0 (occurs when n=0), k is still a multiple of 9. I also got stuck for a bit on answer choice E.) "K is a multiple of 6" until some plug-n-chug at n=1 disproved this answer.

I'd be very interested in seeing how other people solved this - please post if you used a different route of thinking.

D says that k has to be a multiple of 9
K = sum of the digits of M

so lets a couple of cases

we know m = 18 n

when n=0 m=0 so k =0 and k is a multiple of 9 , D is true

when n=1,2...6 m= 18, 36,....108 so k = 9 again K is a multiple of 9 , D is again true

when n = -1 or -2 or -6 then m = -18 or -36 or -108 then k = 7 or 3 or 7 ..but now K is not a multiple of 9 ??

so how can D always be true ??

Please note question does not mention that n is a positive integer or M is a positive integer .
if n is a negative integer as shown above then m will be negative and the sum of the digits of M will not always be 9 so please do explain
how D is always true ??

The sum of the digits of -18 is still 9 (1+8) not not 7 (-1+8).

Hope it's clear.
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Bunuel wrote:
stne wrote:
Club909 wrote:
Great question. Made me think for a while.

My answer is D.) "K is a multiple of 9".

We know that M is always divisible by 18 which means that M is always divisible by 9. This implies that the sum of the digits of M (also referred to as k) will ALWAYs be divisible by 9 (refer to the divisibility rules if you want confirmation).

I kept thinking 0 was a loophole until I realized that 0 is a multiple of ALL integers so in the case k=0 (occurs when n=0), k is still a multiple of 9. I also got stuck for a bit on answer choice E.) "K is a multiple of 6" until some plug-n-chug at n=1 disproved this answer.

I'd be very interested in seeing how other people solved this - please post if you used a different route of thinking.

D says that k has to be a multiple of 9
K = sum of the digits of M

so lets a couple of cases

we know m = 18 n

when n=0 m=0 so k =0 and k is a multiple of 9 , D is true

when n=1,2...6 m= 18, 36,....108 so k = 9 again K is a multiple of 9 , D is again true

when n = -1 or -2 or -6 then m = -18 or -36 or -108 then k = 7 or 3 or 7 ..but now K is not a multiple of 9 ??

so how can D always be true ??

Please note question does not mention that n is a positive integer or M is a positive integer .
if n is a negative integer as shown above then m will be negative and the sum of the digits of M will not always be 9 so please do explain
how D is always true ??

The sum of the digits of -18 is still 9 (1+8) not not 7 (-1+8).

Hope it's clear.

Ok, if - 18 = 1+8 then D is always true , Got it
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GMAT 1: 690 Q47 V38 Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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m=18n means 18*n? when I saw 18n, I thought k=m=1+8+n=9+n and couldnt find any solution
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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LalaB wrote:
m=18n means 18*n? when I saw 18n, I thought k=m=1+8+n=9+n and couldnt find any solution

It wasn't stated "the three-digit number 18n".
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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B is incorrect. Try m=18 * 11 and you will find that the sum of digits is not 9.

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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Could someone explain in which cases k is not a multiple of 6. Thank you!
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Stiv wrote:
If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6

Could someone explain in which cases k is not a multiple of 6. Thank you!

m=18 --> k=1+8=9 --> 9 is NOT a multiple of 6.
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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superpus07 wrote:
If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6

I could figure out an easy solution for this one. Anyone have any idea how to solve this efficiently?
Will throw some nice Kudos out there!

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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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Can someone post an example of a case when C is not true?
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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lajulajay wrote:
Can someone post an example of a case when C is not true?

Hello lajulajay

Let try n = 11
==> m = 18*11 = 198
==> k = 1 + 9 + 8 = 18
==> 2k = 36

But 198 / 36 = 5.5 ==> C is not always correct.

Hope it helps.
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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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superpus07 wrote:
If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6

We can apply the concept of “Digital Root” here.

Digital root is consecutive summation of digits of a number until the sum reaches a one digit value. Although the question is not asking about consecutive summation, all the principles of digital root can still be applied here.

There is ,so called, “Rule of 9”. That is when we multiple any number by 9 its digital root will ALWAYS be 9. Also from the perspective of divisibility rules: when we multiply any number by 9 we’ll make this number a multiple of 9, thus sum of its digit will be divisible by 9.

We have $$m=18n$$ ---> $$m=2n*9$$

Whatever nonzero $$n$$ we plug in the digital root of $$m$$ will ALWAYS be 9.

Now, because we are not asked about digital root directly we need to know another important quality of digital root. Digital root of ANY number has a cycle of 9.

$$9; 18=1+8=9; 27=2+7=9; 36=3+6=9 …. 1152 = 1+1+5+2 = 9 …$$

So we have arithmetic progression of multiples of 9.

Hence if we multiply ANY number by 9, sum of its digit will ALWAYS be a multiple of 9. Even if we plug in n=0, we get m=0 and 0 is multiple of any number.

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Re: If k is the sum of the digits of integer m, and m=18n, where  [#permalink]

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superpus07 wrote:
If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6

Its an easy problem...lets look at this way.. as per question $$\frac{m}{18} = n$$.. which means $$m = 18p$$ where $$p = 0,1,2,3,4,5,...$$.
Now, K = sum of digits of m, however when m = 0, sum of digits of K will not be equal to 9.
When, m =18.. K will not be multiple of 6...
K will always be multiple of 9.
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