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.


By taking,’ a’ as any positive integer and b = 3.

Applying Euclid’s algorithm


a = 3q + r


Here,


So, a = 3q or 3q+1 or 3q+2


And,


a2 = (3q)2 or (3q+1)2 or (3q+2)2


a2 = (9q2) or 9q2+6q+1 or 9q2+12q+4


a2 = 3(3q2)2 or (3q2 + 2q)+1 or 3(3q2+4q+1)+1


a2 = 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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