Question

Prove that square of an positive integer os of the form 4n or 4n+1 for some integer n

Answer 1

Let a be any positive integer.

a=bn+r
a=4n+r

0≤r<4

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

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

Square of any positive integer:-

1²=1=4(0)+1 (r=1)
2²=4=4(1)+0 (r=0)
3²=9=4(2)+1 (r=1)
4²=16=4(4)+0 (r=0)

Therefore,the square of any positive integer is of the form 4n or 4n+1 for any integer n.

Hence proved