Number Properties | OG 12

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If n is a positive integer, is n^3 - n divisible by 4 ?

(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6.

Here, if we look at statement 1 then K can also be 0 which is a positive integer and in that case n=1, Hence n= odd and statement 1 is not sufficient but it is given suff. in the explanation since it does not considers the possibility of k=0. Please explain!

[Achilles_heel]

Hi nishatfarhat87,

with your logic, the statement one will still be sufficient as in that case n-1 will be 0 and n^3 - n will be 0 as well. since 0 is divisible by every integer, the sufficiency is proved

[Brent@GMATPrepNow]

If n is a positive integer, is n^3 - n divisible by 4 ?
(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6.

Target question: Is n^3 - n divisible by 4?

This is a great candidate for rephrasing the target question.

Aside: Rephrasing the target question can often make data sufficiency questions easier (and faster) to solve. We have a free video on this strategy: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Notice that we can take n^3 - n and factor it to get n(n^2 - 1), which equals n(n-1)(n+1) or (n-1)(n)(n+1)
Now recognize that n-1, n, and n+1 are three consecutive integers. The GMAT often hides this kind of information within given algebraic expressions.

So, at this point, we can rephrase the target question as: Is the product of 3 consecutive integers divisible by 4?

If we dig a little deeper, we can further rephrase the target question to make the question even easier to solve.
To do this, we'll ask, "Under what circumstances is the product of 3 consecutive integers divisible by 4? Well, there are two such circumstances.
Circumstance 1: The first and last integers are even. For example, the product of 2, 3, and 4 will be divisible by 4. In this circumstance, the middle number (n) is odd.
Circumstance 2: The middle integer is divisible by 4. For example, the product of 7, 8, and 9 must be divisible by 4 since the number 8 is already divisible by 4. In this circumstance, the middle number (n) is divisible by 4.

Given these two circumstances, we can rephrase the target question as: Is n either odd or divisible by 4?

At this point, we can check the statements.

Statement 1: n = 2k + 1, where k is an integer
This is a very clever way of telling us that n is odd. In fact, this is the formal definition of an odd number.
Since n is odd, we can now answer the rephrased target question with certainty.
So, statement 1 is SUFFICIENT

Statement 2: n^2 + n is divisible by 6
Notice that we can take n^2 + n and rewrite it as (n)(n+1), and we know that n and n+1 are two consecutive integers.
This information yields different possible cases, here are two.
case a: n=2, n+1=3, in which case n is neither odd nor divisible by 4
case b: n=3, n+1=4, in which case n is odd
Since statement 2 yields conflicting answers to our rephrased target question, it is NOT SUFFICIENT.

Answer = A

Cheers,
Brent