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# If m and k are non-zero integers, is m a multiple of k?

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Math Expert
Joined: 02 Sep 2009
Posts: 61396
If m and k are non-zero integers, is m a multiple of k?  [#permalink]

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16 Dec 2019, 01:43
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45% (medium)

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64% (01:43) correct 36% (01:51) wrong based on 55 sessions

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If m and k are non-zero integers, is m a multiple of k?

(1) $$\frac{m^2 + m}{k}$$ is an integer.

(2) $$m = 2k^2 − 3k$$

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Joined: 16 Jan 2019
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Concentration: General Management
WE: Sales (Other)
If m and k are non-zero integers, is m a multiple of k?  [#permalink]

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16 Dec 2019, 03:52
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Bunuel wrote:
If m and k are non-zero integers, is m a multiple of k?

(1) $$\frac{m^2 + m}{k}$$ is an integer.

(2) $$m = 2k^2 − 3k$$

Are You Up For the Challenge: 700 Level Questions

(1) $$\frac{m^2 + m}{k}$$ is an integer.

$$\frac{m(m+1)}{k}$$ is an integer

So either $$m$$ is a multiple of $$k$$ or $$m+1$$ is a multiple of $$k$$

Not sufficient

(2) $$m = 2k^2 − 3k$$

$$m=k(2k-3)$$

Therefore, $$m$$ is a multiple of $$k$$

Sufficient

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Joined: 08 Aug 2014
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GMAT Date: 10-02-2015
Re: If m and k are non-zero integers, is m a multiple of k?  [#permalink]

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16 Dec 2019, 04:05
I will try to explain the way I understand and solved it. Please pardon me if it's not clear to all.

Statement 1 is clearly insufficient.

If we try to open the expression, we achieve something like

m^2 + m = Kn (where n is an integer value)

m (m+1)= kn.
The values of the expression could be

m= kn, or m+1= kn, so it's practically hard to tell which of them is the multiple. it could be either m or m+1

We can confirm this by plugging in real numbers to check the initial expression.

Let us use M= 4 for our example
4^2 +4 =20.

Our K can be either 2, 4, 5 , 10 or 20 because only these numbers can divide 20 without leaving any remainder.

If K= 5 or 10, then m= 4 is not a multiple of K.

But if k= 4 or 20, then m= 4 becomes a multiple.

So statement 1 clearly is insufficient.

STATEMENT 2: M= 2k^2 - 3k

We can factorise the expression to obtain something like

M = k (2K-3)

We can then subsequently replace the (2k-3) with

Therefore m= kn.

To further confirm, you could plug in real numbers and seethat the expression is clearly correct .

Statement 2 is sufficient

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Joined: 12 Dec 2019
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If m and k are non-zero integers, is m a multiple of k?  [#permalink]

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16 Dec 2019, 10:35
Rephrasing the question stem: Is m divisible by k?

statement 1) (m^2+ m ) / k
----> m (m+1) is divisible by K
----> either m or (m+1) or both m &(m+1) is/are divisible by k
----> not sufficient

statement 2) m = 2k^2 - 3k
----> m = k(2k-3)
----> since k is an integer ---> 2k-3 is an integer ----> m = k . Integer
----> sufficient

If m and k are non-zero integers, is m a multiple of k?   [#permalink] 16 Dec 2019, 10:35
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