NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 1: Real Numbers - Exercise 1.1 - Multiple Choice Questions

Question 3

Let \(n\) be an odd integer, so we can write \(n = 2k+1\) for some integer \(k\).

Now, \(n^2 - 1 = (2k+1)^2 - 1 = 4k(k+1)\).

Since \(k\) and \(k+1\) are consecutive numbers, one of them is always even. Therefore, their product \(k(k+1)\) is divisible by 2.

That makes the whole expression divisible by \(4 \times 2 = 8\).

So, \(n^2 - 1\) is divisible by 8 when \(n\) is odd.