Question

n² – 1 is divisible by 8, if n is  

(A) an integer 

(B) a natural number

(C) an odd integer 

(D) an even integer

Answer 1

(C) 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,

⇒ x = (2k)²-1

⇒ 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,

⇒ x = 2k + 1

⇒ x = (2k+1)² – 1

⇒ x = 4k² + 4k + 1 – 1

⇒ x = 4k² + 4k

⇒ 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) = 4 which is divisible by 8 .

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

From the above two observation, we can conclude that, if n is odd, if n odd, n²-1 is divisible by 8.

Hence, option (C) is the correct answer.