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

(hint : let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.)
Ans. let a be any positive integer and b = 3.
Then a = 3q + r for some integer q ≥ 0
And r = 0, 1, 2 because 0 ≤ r < 3 Or, a = (3q)² or (3q +1)² or (3q + 2)² a =(9q²) or 9q² + 6q + 1 or 9q² + 12q + 4 = 3 x (3q³) or 3 (3q² +2q) + 1 or 3 (3q² + 4q + 1) + 1 = 3k₁ or 3k₂ + 1 or 3k₂ + 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. (hint : let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.) Ans. let a be any positive integer and b = 3. Then a = 3q + r for some integer q ≥ 0 And r = 0, 1, 2 because 0 ≤ r < 3 Or, a = (3q)² or (3q +1)² or (3q + 2)² a =(9q²) or 9q² + 6q + 1 or 9q² + 12q + 4 = 3 x (3q³) or 3 (3q² +2q) + 1 or 3 (3q² + 4q + 1) + 1 = 3k₁ or 3k₂ + 1 or 3k₂ + 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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