For any integer n, which of the following expressions must be divisible by 2? — GMAT

For any integer $n$ , which of the following expressions must be divisible by $2$ ?

The expression

n^2 + n

is always divisible by

2

for any integer

n

The expression

n^2 + n

factors into

n(n + 1)

. Because

n

and

n + 1

are consecutive integers, one of them must be even. Any integer multiplied by an even integer produces an even number, guaranteeing that

n^2 + n

is divisible by

2

for every integer

n

Factor the given algebraic expression.

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

Factoring out

n

reveals the product of two consecutive integers.

Analyze the parity of consecutive integers

n

and

n + 1

In any pair of consecutive integers

(n, n + 1)

, exactly one number is even.

Even and odd integers alternate sequentially.

Determine divisibility by

2

Because one of the factors is even (divisible by

2

), their product

n(n + 1)

is always divisible by

2

Any integer multiplied by an even number yields an even result.

The product of any two consecutive integers is always even and therefore divisible by 2.

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