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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

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⇒ No , square of any positive integer of the form 3m + 1 is always in the form in the 3m + 1 , but it'snot in the form of - Either 3m or 3m + 2 because of the following solution :

⇒ a = bq + r Let ''a'' be any positive integer and 'q be the quotient and let ''r'' be the remainder .

Therefore we get ,

a = ( 3 q + 1 )² { We square it , as according to the question }

a = 9 q² + 6 q² + 1² { (a + b )² = a² + b² + 2 a b }

a = 3 ( 3 q² + 2 q²) + 1

a = 3 q + 1 ( Where , q is 3 q + 1 )

So, we got that it is always in the form of 3 q + 1.