Question

Show that the square of any positive integer may written as 3m or 3m + 1, where m is some integer.

Answer 1

Answer:

Step-by-step explanation:

Let us take  any positive integer  b = 3.  

Then using Euclid's algorithm we get a = 3m + r

( r is the remainder and value of m is more than or equal to 0 and r = 0, 1, 2 because 0 ≤ r < b and the value of b is 3 So the  possible values will  be 3m+0 , 3m+1 and 3m+2  )

(3q)² = 9q²

        = 3 x (3q²)

         =3m   ( m = 3q²  )

(3q+1)²  = (3q)² + (2 x 3q x 1 ) + 1²

             =9q² + 6q + 1

              = 3(3q² + 2q) + 1  

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

(3q+2)²  = (3q)² + (2 x 3q x 2 ) + 2²  

             =9q² + 12q +4

             = 3(3q² + 4q +1) + 1

             =3m+ 1  (m= 3q² + 4q + 1  )

So Square of any positive integer is either of the form 3m or 3m + 1 for some integer m.