Question

Prove that the square of any positive integer is either of the form 5q or 5q + 1 or 5q+4 for some positive integer q

Answer 1

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SOLUTION:
        
               Let a be the positive even integer then a is form of 5p
                 2   
               a   =  (5p) square 
                    =  25p square
                    = 5(5p square) {where 5p square = q}
                    =5q

 ---------------  if a is an odd number then a is in form of 5p+1
               2             
             a    = (5p+1 )square
                   = 25p square +1
                   =5(5p square )+1 [where 5p square = q]
                   = 5q  +1
--------------if a is a od number in the form 5p+2
                    2 
                 a     = (5p+2) square 
                        = 25p square +4
                        = 5(5p square)+4  [where 5p square =q]
                        = 5q +4
hence proved

Answer 2

Answer:

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