**Question**

**(from MGMAT Practice Test #1)**

For positive integer

*k*, is the expression (

*k*+ 2)(

*k*

^{2}+ 4

*k*+ 3) divisible by 4?

(1)

*k*is divisible by 8.

(2) | k + 13 | is an odd integer. |

**Solution**

The quadratic expression

*k*

^{2}+ 4

*k*+ 3 can be factored to yield (

*k*+ 1)(

*k*+ 3). Thus, the expression in the question stem can be restated as (

*k*+ 1)(

*k*+ 2)(

*k*+ 3), or the product of three consecutive integers. This product will be divisible by 4 if one of two conditions are met:

If

*k*is odd, both

*k*+ 1 and

*k*+ 3 must be even, and the product (

*k*+ 1)(

*k*+ 2)(

*k*+ 3) would be divisible by 2 twice. Therefore, if

*k*is odd, our product must be divisible by 4.

If

*k*is even, both

*k*+ 1 and

*k*+ 3 must be odd, and the product (

*k*+ 1)(

*k*+ 2)(

*k*+ 3) would be divisible by 4 only if

*k*+ 2, the only even integer among the three, were itself divisible by 4.

The question might therefore be rephrased “Is

*k*odd, OR is

*k*+ 2 divisible by 4?” Note that a ‘yes’ to either of the conditions would suffice, but to answer 'no' to the question would require a ‘no’ to both conditions.

(1) SUFFICIENT: If

*k*is divisible by 8, it must be both even and divisible by 4. If

*k*is divisible by 4,

*k*+ 2 cannot be divisible by 4. Therefore, statement (1) yields a definitive ‘no’ to both conditions in our rephrased question;

*k*is not odd, and

*k*+ 2 is not divisible by 4.

(2) INSUFFICIENT: If

*k*+ 1 is divisible by 3,

*k*+ 1 must be an odd integer, and

*k*an even integer. However, we do not have sufficient information to determine whether

*k*or

*k*+ 2 is divisible by 4.

The correct answer is A.

**Query**

I think the answer is D. In Statement (2), for (k+1)/3 to be an odd INTEGER, k+1 has to be divisible by 3. Also, k must be even. Therefore, k could be 2,8,... And if k is an even integer, (

*k*+ 1)(

*k*+ 2)(

*k*+ 3) will not be divisible by 4 for any of these values of k. Therefore, Statement (2) also gives us the answer. Please tell me where I'm going wrong.

## 1 comment:

Put 2 as a solution for k in (k+1)(k+2)(k+3). This equation is this divisible by 4. For other solutions, k=8,14 etc, this is not the case. Hence, (2) is Insufficient.

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