The remainder obtained when `51^25` is divided by 13 is

Options

A. 3
B. 7
C. 12
D. 11

Answer

To find the remainder when \( 51^{25} \) is divided by 13, we can follow these steps:

### Step 1: Simplify the Base
First, we simplify \( 51 \) modulo \( 13 \):
\[ 51 \div 13 = 3 \quad \text{(quotient)} \]
\[ 51 - (3 \times 13) = 51 - 39 = 12 \]
Thus, \( 51 \equiv 12 \mod 13 \).

### Step 2: Rewrite the Expression
Now we can rewrite the expression \( 51^{25} \) as:
\[ 51^{25} \equiv 12^{25} \mod 13 \]

### Step 3: Apply 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, \( p = 13 \) and \( a = 12 \). Since \( 12 \) is not divisible by \( 13 \), we can apply the theorem:
\[ 12^{12} \equiv 1 \mod 13 \]

### Step 4: Reduce the Exponent
Now, we need to reduce the exponent \( 25 \) modulo \( 12 \) (since \( 12^{12} \equiv 1 \)):
\[ 25 \div 12 = 2 \quad \text{(quotient)} \]
\[ 25 - (2 \times 12) = 25 - 24 = 1 \]
Thus, \( 25 \equiv 1 \mod 12 \).

### Step 5: Calculate the Remainder
Now we can simplify \( 12^{25} \) using the reduced exponent:
\[ 12^{25} \equiv 12^{1} \mod 13 \]
\[ 12^{1} \equiv 12 \mod 13 \]

### Step 6: Conclusion
The remainder when \( 51^{25} \) is divided by \( 13 \) is:
\[ \boxed{12} \]