Find the remainder when `11^(12)` is divided by 7.

Options

A. 0
B. 1
C. 3
D. 5

Explanation

To find the remainder when \( 11^{12} \) is divided by 7, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Simplify \( 11 \mod 7 \)
First, we find the equivalent of \( 11 \) modulo \( 7 \):
\[ 11 \div 7 = 1 \quad \text{(remainder 4)} \]
So,
\[ 11 \equiv 4 \mod 7 \]

### Step 2: Rewrite the expression using the equivalent
Now we can rewrite \( 11^{12} \) in terms of \( 4 \):
\[ 11^{12} \equiv 4^{12} \mod 7 \]

### Step 3: Use Fermat's Little Theorem
Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:
\[ a^{p-1} \equiv 1 \mod p \]
Here, \( p = 7 \) and \( a = 4 \). Since \( 4 \) is not divisible by \( 7 \):
\[ 4^{6} \equiv 1 \mod 7 \]

### Step 4: Reduce the exponent modulo \( 6 \)
Now, we need to reduce \( 12 \) modulo \( 6 \):
\[ 12 \div 6 = 2 \quad \text{(remainder 0)} \]
Thus,
\[ 4^{12} = (4^{6})^2 \equiv 1^2 \equiv 1 \mod 7 \]

### Step 5: Conclusion
Therefore, the remainder when \( 11^{12} \) is divided by \( 7 \) is:
\[ \boxed{1} \]