If \( n \equiv 1 \pmod{3} \), then \( n^2 \equiv 1 \pmod{3} \),
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- Step 1: Check modulo 3
- The possible residues modulo 3 are 0, 1, and 2:
- If \( n \equiv 0 \pmod{3} \), then \( n^2 \equiv 0 \pmod{3} \),
- If \( n \equiv 2 \pmod{3} \), then \( n^2 \equiv 4 \equiv 1 \pmod{3} \).
- Thus, \( n^2 \equiv 1 \pmod{3} \) when \( n \equiv 1 \) or \( 2 \pmod{3} \).
- Step 2: Check modulo 5
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