Competition problem — Number theory

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Number theory Difficulty 5.7 AIME, harder Prove it

5. Prove: When $m>2$ , $0^{2}, 1^{2}, \cdots,(m-1)^{2}$ cannot be a complete residue system modulo $m$ .

Solution

5. Since $j^{2} \equiv (m-j)^{2} \pmod{m}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.