Exercise 502Logic, sets and proofs › Proof techniquesHigh schoolAdded 3 Nov 2026

A parity case analysis

#cases#parity#square

Statement

Prove that the square of any integer has the form $3k$ or $3k+1$ , but never $3k+2$ .

Hint

Consider $n\equiv0,1,2\pmod3$ .

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Solution

Cases.
- $n\equiv0\pmod3$ : $n^2\equiv0\pmod3$ , so $n^2=3k$ .
- $n\equiv1\pmod3$ : $n^2\equiv1\pmod3$ , so $n^2=3k+1$ .
- $n\equiv2\pmod3$ : $n^2\equiv4\equiv1\pmod3$ , so $n^2=3k+1$ .

Conclusion. The square of any integer is $\equiv0$ or $1\pmod3$ , never $2$ .

A parity case analysis

Statement

Solution

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