A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.


No


By Euclid’s Lemma, b = a × q + r, 0 ≤ r < a


Here, b is any positive integer


According to the question, a = 3 b = 3q + r for 0 r < 3


So, this must be in the form 3 q, 3q + 1 or 3 q + 2.


and (3 q + 1)2 = 9q2 + 6q + 1


(3 q + 1)2 = 3 (3q2 + 2q) + 1


(3 q + 1)2 = 3m + 1


[where m = 3q2 + 2q ]


Hence, square of a positive integer of the form 3q + 1 is always in the form 3m + 1 for some integer m.


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