For any integer n, which of the following expressions MUST be an even integer? — GMAT

For any integer $n$ , which of the following expressions MUST be an even integer?

The expression

(n + 1)^3 - (n - 1)^3

MUST be an even integer for all integer values of

n

Simplifying the expression

(n + 1)^3 - (n - 1)^3

yields

6n^2 + 2

, which equals

2(3n^2 + 1)

. Since

3n^2 + 1

is always an integer for any integer

n

, multiplying by 2 guarantees the expression is an even integer for all values of

n

, including zero and negative integers.

Expand both cubic terms algebraically.

(n + 1)^3 = n^3 + 3n^2 + 3n + 1

and

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

Expanding the terms allows combining like terms to simplify the expression.

Subtract the expanded expression

(n - 1)^3

from

(n + 1)^3

(n^3 + 3n^2 + 3n + 1) - (n^3 - 3n^2 + 3n - 1) = 6n^2 + 2

The cubic terms (

n^3

) and linear terms (

3n

) cancel out.

Factor out a common factor of 2 from the simplified expression.

6n^2 + 2 = 2(3n^2 + 1)

Any integer that can be expressed as

2k

, where

k

is an integer, is by definition even. Since

n

is an integer,

3n^2 + 1

is an integer, making

2(3n^2 + 1)

an even integer for all

n

Parity rules for algebraic expressions and consecutive integer products

Instead of algebraic expansion, test values of

n

with different parities, including

n = 0

(even) and

n = 1

(odd). For

n = 0

(0 + 1)^3 - (0 - 1)^3 = 1 - (-1) = 2

(even). For

n = 1

(1 + 1)^3 - (1 - 1)^3 = 8 - 0 = 8

(even). Because

n + 1

and

n - 1

always have the same parity, their cubes also share the same parity, so their difference is always even.

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