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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’. 

(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)

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(i) When n = 3m, 

n2 = 9m2 

= 3 × 3m2 = 3q (∵ q = 3m2

(ii) If n = 3m + 1 

n2 = (3m + 1)2 

= 9m2 + 6m + 1 

= 3 (3m2 + 2m) + 1

= 3q + 1 (∵ 3m2 + 2m = q) 

∴ Square of any positive integer is either of the form 3m or 3m + 1 for some integer ‘m’.

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