Question

Show that square of any odd positive integer is of the form 8n +1 ,where n is any integer.​

Answer 1

Let a be any positive odd integer and b=4

Every odd positive integer of the form of either 4q+1 or 4q+3

Case 1: When a=2q+1,

a²=(4q+1)²

=16q²+8q+1

=8q(2q+1)+1

=8n+1,where n=q(2q+1)

Case 2: When a=2q+3,

a²=(4q+3)²

=16q²+24q+3

=8q(2q+3)+3

=8n+3,where n=2q+3

Thus,square of every positive odd integer is of the form 8n+1 or 8n+3

Hence,proved