Divisibility of 10^n − 1 by 11 For which values of n is (10^n − 1) divisible by 11?

Introduction / Context:
Understanding periodic behavior modulo 11 is essential in number theory. This question explores when numbers of the form 10^n − 1 are divisible by 11.

Given Data / Assumptions:

• We analyze 10^n modulo 11.
• Use modular exponent properties.
• n is a positive integer (including n = 0 is trivial but not required).

Concept / Approach:
Since 10 ≡ −1 (mod 11), we have 10^n ≡ (−1)^n (mod 11). Therefore 10^n − 1 ≡ (−1)^n − 1. This equals 0 exactly when n is even (because (−1)^{even} = 1).

Step-by-Step Solution:
Compute 10 ≡ −1 (mod 11).Then 10^n ≡ (−1)^n (mod 11).So 10^n − 1 ≡ (−1)^n − 1.If n is even: (−1)^n = 1 ⇒ expression ≡ 0 ⇒ divisible by 11.If n is odd: (−1)^n = −1 ⇒ expression ≡ −2 ⇒ not divisible by 11.

Verification / Alternative check:
Try n = 2: 10^2 − 1 = 99 divisible by 11. Try n = 1: 9 not divisible by 11. Pattern holds.

Why Other Options Are Wrong:
All values / odd / multiples of 11 contradict the modular result based on (−1)^n.

Common Pitfalls:
Assuming divisibility for all n because 99 is divisible by 11; forgetting 10 ≡ −1 (mod 11) and its parity effect.

Final Answer:
Even values of n