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If n is an odd number, which of the following must be even?

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If n is an odd number, which of the following must be even? [#permalink]

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New post 12 Oct 2017, 23:02
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If n is an odd number, which of the following must be even? [#permalink]

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New post 13 Oct 2017, 02:34
Bunuel wrote:
If n is an odd number, which of the following must be even?

A. (n - 1)/2

B. (n + 1)/2

C. n^2 + 2n

D. 2n + 2

E. 3n^2 - 2n

Use n = 1, n = 3, and in one case, n= 5

If n is odd, the most likely "must be even" expressions will have terms that are (Even)*(Odd) and/or (Odd + Odd).

Scanning the choices, "2n" in Answer D should stand out.

To guarantee that a term is even, we use 2n. 2 is even.
(Even) * (Even OR Odd) = Even.

Answer (D) 2n + 2
Even * Odd = Even, then
Even + Even = Even CORRECT

Nothing will make that expression odd. Examples:
n = 1: (2)(1) + 2 = (2 + 2) = 4
n = 3: (2)(3) + 2 = (6 + 2) = 8
n = -1: (2)(-1) + 2 = (-2+2) = 0

Checking the others:

A. (n - 1)/2
Division is unpredictable.
n = 1 yields result of 0, even.
BUT
If n = 3: (3-1)/2 =(2/2) = 1. REJECT

B. (n + 1)/2
Again, unpredictable.
n = 3: 4/2 = 2. Even. BUT
n = 5: 6/2 = 3. Odd. REJECT

C. n^2 + 2n
(Odd * Odd) = Odd, then
Odd + Even = Odd.

An odd number, squared, is always odd. 2n is even.
n = 1: (1 + 2) = 3. Odd. REJECT

E. 3n^2 - 2n
Odd * (Odd*Odd) = Odd, then
Odd - Even = Odd
n = 1: (3*1) - 2 = (3-2) = 1. REJECT

Answer
[Reveal] Spoiler:
D

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If n is an odd number, which of the following must be even?   [#permalink] 13 Oct 2017, 02:34
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If n is an odd number, which of the following must be even?

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