If n is a negative odd integer, which of the following expressions MUST be a positive even… — GRE General Test

If $n$ is a negative odd integer, which of the following expressions MUST be a positive even integer?

The expression

n^2 + 1

must be a positive even integer.

Squaring any negative odd integer

n

produces a positive odd integer (

n^2 \ge 1

). Adding

1

to a positive odd integer always yields a positive even integer (

n^2 + 1 \ge 2

Analyze the parity and sign of

n^2

Since

n

is negative and odd, any even power of

n

(such as

n^2

) produces a positive odd integer.

Negative times negative yields positive, and odd times odd yields odd.

Analyze the effect of adding

1

n^2

n^2 + 1

is the sum of a positive odd integer and

1

, which results in a positive even integer.

Adding

1

to any odd integer results in an even integer, and adding

1

to a positive integer maintains positivity.

Parity and sign rules for exponents and addition

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