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# For positive integer k, is the expression (k + 2)(k2 + 4k +

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For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 11:53
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For positive integer k, is the expression (k + 2)(k^2 + 4k + 3) divisible by 4?

(1) k is divisible by 8.
(2) (k + 1)/3 is an odd integer.

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-positive-integer-k-is-the-expression-k-2-k-2-4k-84196.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Aug 2014, 06:18, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 12:01
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(k+2)(k+3)(k+1).

1) SUFF.
if k is divisible by 8 => k+2 is 2 in mod 4; k+3 = 3 in mod 4 and k+1=1 in mod 4.
So this multiplication becomes 2x3x1= 6 = 2 in mod 4. Not divisible.

2)INSUFF

(k+1)/3=odd, so k+1 = odd so k is even.
if k mod 4 = 0 then this eq. is not divisible by 4.
if k mod 4 =2 then eq. is divisible by 4.
So we can not know.

A

Last edited by maliyeci on 30 Sep 2009, 13:15, edited 1 time in total.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 12:04
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case 1)
k=8n, where n is any positive integer.

f(k)= (8n + 2) * ( (8n)^2 + 32n + 3)
2 * (4n + 1) * ( even + even + odd )
2 * odd * odd = > not divisible by 4

sufficient condition

case 2)

(k+1)/3 = odd integer = 2n+1
k=6n+2=even integer.

f(k)= (6n + 4) * ( even + even + odd ) = 2 * (3n+2) * odd

3n+2 can be even and odd, so Insufficient
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Last edited by sudeep on 30 Sep 2009, 12:17, edited 2 times in total.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 12:06
amitgovin wrote:
For positive integer k, is the expression (k + 2)(k2 + 4k + 3) divisible by 4?

(1) k is divisible by 8.

(2)
(K + 1)/3 is an odd integer.

IMO A.
$$(k + 2)(k2 + 4k + 3) = 6K^2+12K+6$$

Statement 1: as K ismultiple of 8, $$6K^2+12K$$ is divisible by 8. So $$6K^2+12K+6$$ is NOT divisible by 8.
Sufficient.
Statement 2: As $$(K+1)/3$$ is an odd integer, K must be a multiple of 3.
Insufficient.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 12:09
Aargh! sorry, this question has been frustrating me. I understand 1) however I disagree with 2).

The stem says that K is a positive integer. Given that and the fact that we know that K is even from stem 2) we know that K is at MINIMUM 2. This is where I seem to disagree with your answer (which is apparently the correct one according to the MGMAT). IF k is at minimum 2, then K+2 is 4, thus K+2 is divisible by 4 and that makes the entire expression div by 4.

Tell me why I'm wrong. thanks.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 12:22
I got it completely WRONG!!!

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 13:15
amitgovin wrote:
Aargh! sorry, this question has been frustrating me. I understand 1) however I disagree with 2).

The stem says that K is a positive integer. Given that and the fact that we know that K is even from stem 2) we know that K is at MINIMUM 2. This is where I seem to disagree with your answer (which is apparently the correct one according to the MGMAT). IF k is at minimum 2, then K+2 is 4, thus K+2 is divisible by 4 and that makes the entire expression div by 4.

Tell me why I'm wrong. thanks.

Because there are possibilities other than 2 and they are not divisible. For example 8. But there are possibilites other than 2 and they are divisible. For example 14

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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30 Sep 2009, 20:11
amitgovin wrote:
Aargh! sorry, this question has been frustrating me. I understand 1) however I disagree with 2).

The stem says that K is a positive integer. Given that and the fact that we know that K is even from stem 2) we know that K is at MINIMUM 2. This is where I seem to disagree with your answer (which is apparently the correct one according to the MGMAT). IF k is at minimum 2, then K+2 is 4, thus K+2 is divisible by 4 and that makes the entire expression div by 4.

Tell me why I'm wrong. thanks.

Hello amitgovin, hwr are ya? well, i can tell you why you are wrong. you made the honest mistake of thinking k+2 is divisible by 4. It has to be k x 2 for it to be divisible by 4. if you test k = 4 in k + 2, you'd get 4 + 2 = 6 and 6 is not divisible by 4. Again if u test k = 6 in k + 2, you'd get 8 which is divisible by 4. So statement 2 is INSUFF. gud?

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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12 Apr 2010, 12:52
For a product of 2 terms to be div by 4; either product of the factors should be div by 4 (e.g 2x2), or either term should be 0 or any one of the terms should be divisible by 4.

1. K is div by 8:
This means that k is even. Thus, the second term (K^2+4k+3) will always be odd and never 0, thus not being div by 4. K being div by 8 also means K is divisible by 4. If we add 2 to any number, whether negative or positive, divisible by 4, the resulting number will not be divisible by 4. Hence, the second term is not div by 4 either.
Another scenario is k=0 (0 is div by 8). In that case the(k+2)(K^2+4k+3)= 6 i.e. not div by 4.

Hence statement 1 is sufficient (the term is not div by 4)

2. (k+1)/3 is an odd integer. This means, k is even and k+1 is div by 3. if k=8 or 32, neither (k+2) nor (K^2+4K+3) is div by 4. If k=-34 or 2; (K+2) is div by 4. So, dual cases--hence statement 2 is insufficient.

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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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24 Aug 2014, 05:17
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Re: For positive integer k, is the expression (k + 2)(k2 + 4k + [#permalink]

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24 Aug 2014, 06:19
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amitgovin wrote:
For positive integer k, is the expression (k + 2)(k^2 + 4k + 3) divisible by 4?

(1) k is divisible by 8.
(2) (k + 1)/3 is an odd integer.

For positive integer k, is the expression (k + 2)(k^2 + 4k + 3) divisible by 4?

$$(k + 2)(k^2 + 4k + 3)=(k+1)(k+2)(k+3)$$, so the expression is the product of three consecutive integers.

(1) k is divisible by 8 --> $$k=8n=even$$ --> $$(k+1)(k+2)(k+3)=odd*even*odd$$. Now, $$k+2=8n+2$$, though even, is not a multiple of 4 (it's 2 greater than a multiple of 8), therefore the expression is not divisible by 4. Sufficient.

(2) (k + 1)/3 is an odd integer --> $$k+1=3*odd=odd$$ --> $$k=even$$ --> $$(k+1)(k+2)(k+3)=odd*even*odd$$. Now, $$k+2=even$$ may or may not be divisible by 8, therefore the expression may or may not be divisible by 8. For example, consider $$k=2$$ and $$k=6$$. Not sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-positive-integer-k-is-the-expression-k-2-k-2-4k-84196.html
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Re: For positive integer k, is the expression (k + 2)(k2 + 4k +   [#permalink] 24 Aug 2014, 06:19
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