Question

n² – 1 is divisible by 8, if n is ______.

Options

• An integer

• A natural number

• An odd integer

• An even integer

Answer 1

n² – 1 is divisible by 8, if n is an odd integer.

Explanation:

Let x = n² – 1

In the above equation, n can be either even or odd.

Let us assume that n = even.

So, when n = even i.e., n = 2k

Where k is an integer

We get,

`\implies` x = (2k)² – 1

`\implies` x = 4k² – 1

At k = – 1,

x = 4(–1)² – 1

= 4 – 1

= 3, is not divisible by 8.

At k = 0,

x = 4(0)² – 1

= 0 – 1

= – 1, is not divisible by 8

Let us assume that n = odd:

So, when n = odd

i.e., n = 2k + 1

Where k is an integer

We get,

`\implies` x = 2k + 1

`\implies` x = (2k + 1)² – 1

`\implies` x = 4k² + 4k + 1 – 1

`\implies` x = 4k² + 4k

`\implies` x = 4k(k + 1)

At k = –1, x = 4(–1)(–1 + 1) = 0 which is divisible by 8.

At k = 0, x = 4(0)(0 + 1) = 0 which is divisible by 8.

At k = 1, x = 4(1)(1 + 1) = 8 which is divisible by 8.

From the above two observation

We can conclude that, if n is odd, n² – 1 is divisible by 8.