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 of \( 51^{25} \) when divided by 13, we can simplify the problem using modular arithmetic.

### Step-by-Step Solution:

1. **Reduce the base modulo 13**:
\[ 51 \mod 13 \]
To find \( 51 \mod 13 \), we divide 51 by 13:
\[ 51 = 13 \times 3 + 12 \quad \text{(since } 13 \times 3 = 39\text{)} \]
Thus,
\[ 51 \equiv 12 \mod 13 \]

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

3. **Use Fermat's Little Theorem**:
According to Fermat's Little Theorem, if \( p \) is a prime 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:
\[ 12^{12} \equiv 1 \mod 13 \]

4. **Reduce the exponent modulo 12**:
Now we need to reduce the exponent 25 modulo 12:
\[ 25 \mod 12 \]
Dividing 25 by 12 gives:
\[ 25 = 12 \times 2 + 1 \]
Thus,
\[ 25 \equiv 1 \mod 12 \]

5. **Calculate \( 12^{25} \mod 13 \)**:
Now we can substitute back:
\[ 12^{25} \equiv 12^{1} \mod 13 \]
Therefore,
\[ 12^{25} \equiv 12 \mod 13 \]

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