n is a pesitive odd intcger, her.Show,"that n., is divisible byĺo

1.	n is a pesitive odd intcger, her.Show,"that n., is divisible byĺou 8
Answer» Any odd positive integerncan be written in form of 4q+ 1 or 4q+ 3. Ifn= 4q+ 1, Thenn^2- 1 = (4q+ 1)^2- 1 = 16q^2+ 8q+ 1 - 1 = 8q(2q+ 1)which is divisible by 8. Ifn= 4q+ 3, Thenn^2- 1 = (4q+ 3)^2- 1 = 16q^2+ 24q+ 9 - 1 = 8(2q^2+ 3q+ 1) which is divisible by 8. So, it is clear thatn^2- 1 is divisible by 8, if n is an odd positive integer.

n is a pesitive odd intcger, her.Show,"that n., is divisible byĺou 8

Answer»

Any odd positive integerncan be written in form of 4q+ 1 or 4q+ 3.

Ifn= 4q+ 1, Thenn^2- 1 = (4q+ 1)^2- 1 = 16q^2+ 8q+ 1 - 1 = 8q(2q+ 1)which is divisible by 8.

Ifn= 4q+ 3, Thenn^2- 1 = (4q+ 3)^2- 1 = 16q^2+ 24q+ 9 - 1 = 8(2q^2+ 3q+ 1) which is divisible by 8.

So, it is clear thatn^2- 1 is divisible by 8, if n is an odd positive integer.