Question

show that the square of any positive intiger is either of the for 3m or 3m+1 for some intiger

Answer 1

let us take, 'x'= 3q , 3q+1, 3q+2
when, x=3q
        x2 =  (3q) 2
         x2 = 9q2 
        x2  = 3(3q2)
we see that 3q2= m
so we have done the first equation 3m

when , x=3q+1
           x2= (3q+1)2
                                 [since, (a+b)2 = a2+2ab+b2]
           x2= 9q+6q+1
           x2= 3(3q+2q)+1
in this we see that 3q+2q= m
    therefore, this satisfy the equation m+1

Answer 2

We know that a=bq+r
Now if b = 3
a = 3q + r
r = 0,1 or 2
r = 0, a = 3q
a^2 = (3q)^2
= 9q^2
or 3(3q^2)
or 3m where m = 3q^2.
Similarly, take r = 1 and 2.