The number `10^N-1` is divisible by 11 for :

Options

A. even values of N
B. all values of N
C. odd values of N
D. N must be a multiple of 11

Answer

To determine when the number \(10^N - 1\) is divisible by 11, we can use properties of modular arithmetic and the fact that \(10 \equiv -1 \mod 11\).

### Step-by-Step Solution:

1. **Understanding the expression**: We need to check when \(10^N - 1\) is divisible by 11. This means we need to find \(N\) such that:
\[ 10^N - 1 \equiv 0 \mod 11 \]

2. **Substituting the equivalent**: Since \(10 \equiv -1 \mod 11\), we can substitute:
\[ 10^N \equiv (-1)^N \mod 11 \]

3. **Setting up the equation**: We want to find \(N\) such that:
\[ (-1)^N - 1 \equiv 0 \mod 11 \]
This simplifies to:
\[ (-1)^N \equiv 1 \mod 11 \]

4. **Analyzing the condition**: The expression \((-1)^N \equiv 1\) holds true when \(N\) is an even number. This is because:
- If \(N\) is even, \((-1)^N = 1\).
- If \(N\) is odd, \((-1)^N = -1\).

5. **Conclusion**: Therefore, \(10^N - 1\) is divisible by 11 if \(N\) is an even number.

### Final Answer:
The number \(10^N - 1\) is divisible by 11 for all even values of \(N\).