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

Solution:

Any odd positive integer n can be written as 4q + 1 or 4q + 3.

If n = 4q + 1,

n² - 1 = (4q + 1)² - 1

Using the algebraic identity

(a + b)² = a² + b² + 2ab

= 16q² + 8q + 1 - 1

= 8q(2q + 1), is divisible by 8.

If n = 4q + 3,

n² - 1 = (4q + 3)² - 1

Using the algebraic identity

(a + b)² = a² + b² + 2ab

= 16q² + 24q + 9 - 1

= 8(2q² + 3q + 1), is divisible by 8.

So, from the above equations, it is clear that, if n is an odd positive integer

Therefore, n² - 1 is divisible by 8

✦ Try This: If n is an odd integer, then show that n² - 1 is divisible by 6

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

NCERT Exemplar Class 10 Maths Exercise 1.3 Problem 6

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

Summary:

If n is an odd integer, then n² - 1 is divisible by 8.Hence Proved

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