If n is a Positive Integer, Then n(n+1)(n+2) is GMAT Problem Solving

Question: If n is a positive integer, then n(n+1)(n+2) is

1. even only when n is even
2. even only when n is odd
3. odd whenever n is odd
4. divisible by 3 only when n is odd
5. divisible by 4 whenever n is even

Correct Answer: E

Solution and Explanation
Approach Solution 1:

It is given that n is a positive integer. Accordingly, we need to find what is n(n+1)(n+2) whether it is even or odd.

n(n+1)(n+2) will always be even as n is a +ve integer.

This basically rules out options A, B & C. This is because it states that n(n+1)(n+2) is even only when n is even, n(n+1)(n+2) is even when n is odd and n(n+1)(n+2) is odd when n is odd. At Least one of n, n+1 & n+2 will be even as they are consecutive integers.

Even integers when multiplied with even integers is always even
For example: 2*4 = 8 or 6*10 = 60 Hence, always even

Even integers when multiplied with odd integers is always even
For example: 2*3 = 6 or 5 * 8 = 40 Hence, always even

In case of option D, it can be evaluated that-
Either of n, n+1 & n+2 will always be divisible by 3 till the time n is a +ve integer and they are consecutive integers.

Finally, the last option E is what seems to be correct in the given scenario. It states that the n(n+1)(n+2) is divisible by 4 when n is an even integer and a positive integer as well.

Approach Solution 2:

There is another, rather simple approach to solve the problem in finding the answer for n(n+1)(n+2), given that n is a positive integer.

Accordingly, positive integers can be 2, 3, 4, etc.. whereby 0 is neither positive nor negative.
Further, n(n+1)(n+2) might include a positive integer and whether it is multiplied with an even integer or an odd integer, it would turn out to be even.

This eliminates the options of A, B and C and in case of option D, if the integers are consecutive, it will be divisible by 3.

This implies that n(n+1)(n+2) can be equated with 2*3*4, which is divisible by 4.
Further, is n is equal to 6, then for n(n+1)(n+2) = 6*7*8 which is again divisible by 4.
Thus, option E is the correct answer.

Approach Solution 3:

The problem is easiest to solve by substituting numbers for n. We'll try an odd number first and then an even number.

For an odd number, let’s let n = 1: 1(1+1)(1+2) = 1(2)(3) = 6.
We see than choices A and C can’t be the correct choices. Choice A is false because, while the product is even, n is not even. Choice C is false because, while n is odd, the product is even.

For an even number, let’s let n = 2: 2(2+1)(2+2) = 2(3)(4) = 24.

We see than choices B and D can’t be the correct choices. Choice B is false because, while the product is even, n is not odd. Choice D is false because, while the product is divisible by 3, n is even.

Therefore, the only correct answer is choice E. When n is even, the product is indeed divisible by 4.

“If n is a positive integer, then n(n+1)(n+2) is”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Quantitative Review".To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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