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

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According to MGMAT, the answer to the problem below is A. However, I believe it is D. Since the expression is consecutive set, all we need to establish is whether it contains two factors of 2, which would make it divisible by 4. One way to determine this would be to find out whether k is even or odd. Statement 2 tells us that k is even so it should be sufficient. However, according to MGMAT, statement 2, by itself is insufficient. Can someone please confirm this. Thanks.

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 = odd integer

MGMAT Answer Explanation:

The quadratic expression k2 + 4k + 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.

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(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.

[StaceyKoprince]

Statement 2 is insufficient. It does tell us that k is even, but remember that k by itself is not part of what we are testing. What we are testing is the product of the three consecutive integers AFTER k, namely (k+1)(k+2)(k+3).

If k is even, then the product of the next three consecutive integers may or may not be divisible by 4.
If k = 2, then the product IS divisible by 4. 3*4*5 has 4 (or two 2's) as a factor.
If k = 4, then the product IS NOT divisible by 4. 5*6*7 does not have 4 (or two 2's) as a factor - it has only one 2.

It is not enough to know that k is even. We also have to be able to tell that k+2, specifically, is divisible by 4. And statement 2 does not provide enough information to tell.

[richagupta89]

however k cannot equal 4, then (k+1)/3 would not be an odd integer. k could equal 2, then the next integer it can equal based on statement 2, is 8.

[Sage Pearce-Higgins]

I'm not sure I'm totally clear about your question here. According to statement 2, k+1 is an odd integer, so k must be even. Thus k could equal 4.