If \(n\) is a positive integer, which of the following must be even?

[Vincen]

If \(n\) is a positive integer, which of the following must be even?

A. \((n - 1)( n + 1)\)
B. \((n - 2)( n + 1)\)
C. \((n - 2)( n + 4)\)
D. \((n - 3)( n + 1)\)
E. \((n - 3)( n + 5)\)

Answer: B

Source: Princeton Review

[Scott@TargetTestPrep]

If \(n\) is a positive integer, which of the following must be even?

A. \((n - 1)( n + 1)\)
B. \((n - 2)( n + 1)\)
C. \((n - 2)( n + 4)\)
D. \((n - 3)( n + 1)\)
E. \((n - 3)( n + 5)\)

Answer: B

Solution:

Let’s observe the difference between each pair of factors.

A) (n + 1) - (n - 1) = 2

B) (n + 1) - (n - 2) = 3

C) (n + 4) - (n - 2) = 6

D) (n + 1) - (n - 3) = 4

E) (n + 5) - (n - 3) = 8

We see that all the answer choices yield a difference of an even number except choice B. That means for all the answer choices (except B), if one factor is even, the other factor is also even and the product of the two factors will be even. However, if one factor is odd, the other factor is also odd and the product of the two factors will be odd. Therefore, all the answer choices (except B) might not yield an even answer.

Now, let’s look at choice B. Since the difference of the two factors is odd, if one of the factors is odd, the other factor must be even and vice versa. Thus, no matter what, the product must be even.

Answer: B