Prove that if n is odd then 8 divides n^2-1

Given

if n is odd then 8 divides n2-1

Proof

We can proof the given statement directly.

Let n O

Let,

P = n2 – 1

We need to prove that P is divisible by 8

P = n2 – 12

P = (n + 1)(n – 1)

We know that n is odd. This implies that n + 1, n – 1 must be even. Let n – 1 = x

P = x * (x + 2)

Both x and x+2 are even. When we divide x by 2, we have 2 cases

CASE 1: x / 2 = y, y is odd

Divide P by 8

\\*\\*\dfrac{P}{8} = \dfrac{x*(x+2)}{8}\\*\\*\dfrac{P}{8} = \dfrac{y*(x+2)}{4}\\*\\*\dfrac{P}{8} = \dfrac{y*(y+1)}{2}\\*\\*y + 1 \;is\; even.\; Therefore,\; the\; above\; equation\; is\; completely\; divisible\; by\; 8.

CASE 2: x / 2 = y, y is even

\\*\\*\dfrac{P}{8} = \dfrac{x*(x+2)}{8}\\*\\*\dfrac{P}{8} = \dfrac{y*(x+2)}{4}\\*\\*\dfrac{P}{8} = \dfrac{y*(y+1)}{2}\\*\\*Since\; y\; is\; even, \; the\; above\; equation\; is\; completely\; divisible\; by\; 8.\\*\\*

In both the cases, P is completely divisible by 8.

Hence proved.

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