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2014-08-15T04:42:40+05:30

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X² is either  3 m or 3 m + 1          where X >= 1    and m >= 1
let z>=1

1) IF  3 | X X    =>    3 | X      =>      X = 3 z      X² =  3 * (3*z²)  
         THEN   X²  is of form 3m

2) IF  3 does not divide X X   =>  3 does not divide X    =>  X = 3 z+1  or   3z+2

  (i)  X²  = (3z+1)²  =  3 (3z²+2z) + 1  of  the form 3 m + 1

  (ii)  X² =  (3z+2)²  =  3(3z²+4z) + 4 = 3(3z²+4z+1) + 1       
            it is in form  3 m + 1

Hence proved. 
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any natural number, a square or otherwise is in form :  3 m ,  3 m + 1,  3 m + 2
But if it is a square then it is in form : 3 m  or 3m+1 only.

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