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

Options

A. even values of N
B. all values of N
C. odd value 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 analyze the expression step by step.

### Step 1: Understanding the expression
The expression we are analyzing is \( 10^n - 1 \). We want to find out for which values of \( n \) this expression is divisible by 11.

### Step 2: Using the properties of numbers
We can use the fact that \( 10 \equiv -1 \mod{11} \). This means that when we raise 10 to any power \( n \), we can simplify our calculations using this equivalence.

### Step 3: Simplifying the expression
If we substitute \( 10 \equiv -1 \mod{11} \) into our expression, we get:
\[ 10^n \equiv (-1)^n \mod{11} \]
Thus, we can rewrite our expression:
\[ 10^n - 1 \equiv (-1)^n - 1 \mod{11} \]

### Step 4: Analyzing the cases based on \( n \)
Now we need to analyze this expression based on whether \( n \) is even or odd.

- **Case 1: \( n \) is even**
- If \( n \) is even, then \( (-1)^n = 1 \).
- Therefore, \( (-1)^n - 1 = 1 - 1 = 0 \).
- This means \( 10^n - 1 \equiv 0 \mod{11} \), which indicates that \( 10^n - 1 \) is divisible by 11.

- **Case 2: \( n \) is odd**
- If \( n \) is odd, then \( (-1)^n = -1 \).
- Therefore, \( (-1)^n - 1 = -1 - 1 = -2 \).
- This means \( 10^n - 1 \equiv -2 \mod{11} \), which is not equal to 0. Hence, \( 10^n - 1 \) is not divisible by 11.

### Conclusion
From our analysis, we conclude that \( 10^n - 1 \) is divisible by 11 if and only if \( n \) is an even number.

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

---