Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

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Let a be any positive integer and b = 3. Using Euclid’s Division Lemma, a = 3q + r for some integer q ≥ 0 where r = 0, 1, 2 because 0 ≤ r < 3.
Therefore, a = 3q or 3q + 1 or 3q + 2
a² = (3q)² or (3q + 1)² or (3q + 2)²
=(3q)² or 9q² + 6q + 1 or 9q² + 12q + 4
= 3×(3q²) or 3×(3q² + 2q) + 1 or 3×(3q² + 4q + 1) + 1
= 3k₁ or 3k₂ + 1 or 3k₃ + 1
Where k₁, k₂ and k₃ are some positive integers.
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.

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