Question

♡Question!♡
If n is an odd integer, then show that n² - 1 is divisible by 8.

Help !!!!!​

Answer 1

Answer and explanation:

$n {}^{2} - 1 = (n + 1)(n - 1)$

Since n is an odd integers, n+1 and n-1 are even integers.

Let's take the example of 3.

$3 {}^{2} - 1 = 9 - 1 = 8$

Or,

$3 {}^{2} - 1 = (3 + 1)(3 - 1) = 4 \times 2 = 8$

For another example 5,

$5 {}^{2} - 1 = (5 - 1)(5 + 1) = 4 \times 6 = 24$

It can be seen that both n+1 and n-1 are either multiples of 2,or 4.

Therefore, it is natural that n^2 -1 is divisible by 8.

Answer 2

P =2 then, 4P² + 4P = 4(2)² + 4(2) =16 + 8 = 24, it is also 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