Question

If n is an odd integer prove that n^2 - 1 is divisible by 8

Answer 1

Answer:

we know ,

odd number in the form of (2P +1) where P is a natural number ,

so, n² -1 = (2P + 1)² -1

= 4P² + 4P + 1 -1

= 4P² + 4P

now , checking :

P = 1 then,

4P² + 4P = 4(1)² + 4(1) = 4 + 4 = 8 , it is divisible by 8.

P =2 then,

4P² + 4P = 4(2)² + 4(2) =16 + 8 = 24, it is also divisible by 8 .

P =3 then,

4P² + 4P = 4(3)² + 4(3) = 36 + 12 = 48 , divisible by 8

hence, we conclude that 4P² + 4P is divisible by 8 for all natural number .

hence, n² -1 is divisible by 8 for all odd value of n .