Exercise 496Logic, sets and proofs › Proof techniquesHigh schoolAdded 3 Nov 2026

Proof by contraposition

#contrapositive#implication#odd

Statement

Prove: if $n^2$ is odd, then $n$ is odd.

Hint

The contrapositive is: if $n$ is even, then $n^2$ is even.

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Contrapositive. "If $n^2$ odd, then $n$ odd" is equivalent to "if $n$ even, then $n^2$ even".

Direct proof of contrapositive. If $n=2k$ , then $n^2=4k^2=2(2k^2)$ , which is even.

Conclusion. By contraposition, the original statement holds.