Question

$\huge\underline\underline\mathbb\blue{QUESTION}$
If n is an odd integer, then show that n
²– 1 is divisible by 8.​

Answer 1

$\huge\mathbb\fcolorbox{black}{pink}{Answer}$

» We know that,

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

so, n² -1 = (2Q + 1)² -1

= 4Q² + 4Q + 1 -1

$\pink{= 4Q^2\: +\: 4Q}$

now , checking :

Q = 1 then,

4Q² + 4Q = 4(1)² + 4(1) = 4 + 4 = 8 , it is divisible by 8.

Q =2 then,

4Q² + 4Q = 4(2)² + 4(2) =16 + 8 = 24, it is also divisible by 8 .

Q =3 then,

4Q² + 4Q = 4(3)² + 4(3) = 36 + 12 = 48 , divisible by 8

It is concluded that 4Q² + 4Q is divisible by 8 for all natural numbers.

${\blue{\boxed{Hence,\: n^2\: -1 \:is \:divisible\: by\:8 for\: all \:odd\: values\: of\: n.}}}$