Question

show that sguare of any pasitive integer is in the form of 3m or 3m+1

Answer 1

Let any positive integer be a
b=3,then by Euclid's algorithm a=3q+r,for some integer q and we get a=3q ,3q+1,or 3q+2.
Consider
a=3q
Squaring on both side
a square = 9q square
A square = 3(3qsquare) let (3qsquare)be m
a square = 3m
Consider
a=3q+1
Squaring on both sides
a square =(3q+1)whole square
a square=9q square +6q+1
a square =3(3qsquare+2q)+1
Let (3q square+2q) be m
a square=3m+1
Now consider
a=3q+2
Squaring on both side
a square=9q square+6q+4
a square = 3(3q square+2q)+4
a square =3m+4
So, square of positive integer is 3m and 3m+1.

Answer 2

$\textbf{Answer is in Attachment !}$