Question

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

Answer 1

Let a be any positive integer.

a=bm+r
a=3m+r as b=3

0≤r<3
The possible values of r are 0,1 and 2

a=1,2,3,4.....

Square of any positive integer:-

(1)²=1=3(0)+1 (r=1)
(2)²=4=3(1)+1 (r=1)
(3)²=9=3(3)+0 (r=0)
(4)²=16=3(5)+1 (r=1)

Therefore,square of any positive integer is of the form 3m or 3m+1.

Hence proved,

Hope it helps

Answer 2

The question is same you can put 3m +1