n² - 1 is divisible by 8, if n is
a. An integer
b. A natural number
c. An odd integer
d. An even integer

Solution:

Let a = n² - 1
Here n can be even or odd.

When n is even, i.e., n = 2k, where k is an integer

a = (2k)² - 1

a = 4k² - 1.

At k = -1,

a = 4(-1)² - 1 = 4 - 1 = 3,

which is not divisible by 8.

At k = 0,

a = 4(0)² - 1 = 0 - 1 = -1,

which is not divisible by 8.

By assuming that n = odd:

When n = odd

n = 2k + 1, where k is an integer,

So 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) = 0 which is divisible by 8 .

At k = 1,

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

Therefore, if n is odd, then n² - 1 is divisible by 8

✦ Try This: Find the sum of the smallest and the largest 3 digits odd numbers and also prove that it is divisible by 2

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

NCERT Exemplar Class 10 Maths Exercise 1.1 Problem 3

n² - 1 is divisible by 8, if n is a. an integer, b. a natural number, c. an odd integer, d. an even integer

Summary:

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

☛ Related Questions: