When the remainder when `(9)^(11)` is divided by 11.

Options

A. 2
B. 4
C. 7
D. 9

Answer

To find the remainder when \( 9^{11} \) is divided by 11, we can use Fermat's Little Theorem, which states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:

\[ a^{p-1} \equiv 1 \mod p \]

In this case, \( p = 11 \) and \( a = 9 \). Since 9 is not divisible by 11, we can apply the theorem.

### Step 1: Apply Fermat's Little Theorem
According to Fermat's Little Theorem:

\[ 9^{11-1} \equiv 1 \mod 11 \]

This simplifies to:

\[ 9^{10} \equiv 1 \mod 11 \]

### Step 2: Rewrite \( 9^{11} \)
We can express \( 9^{11} \) as:

\[ 9^{11} = 9^{10} \cdot 9 \]

### Step 3: Substitute using the theorem
From Step 1, we know that \( 9^{10} \equiv 1 \mod 11 \). Thus, we can substitute this into our expression:

\[ 9^{11} \equiv 1 \cdot 9 \mod 11 \]

### Step 4: Simplify the expression
This simplifies to:

\[ 9^{11} \equiv 9 \mod 11 \]

### Conclusion
Therefore, the remainder when \( 9^{11} \) is divided by 11 is:

\[ \boxed{9} \]