`(2^(5557))/(13)` find R .

Answer

To find the remainder \( R \) when \( \frac{2^{5557}}{13} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Base and Modulus**: We need to find \( 2^{5557} \mod 13 \).

2. **Use Fermat's Little Theorem**: According to Fermat's Little Theorem, if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:
\[ a^{p-1} \equiv 1 \mod p \]
Here, \( a = 2 \) and \( p = 13 \). Since 2 is not divisible by 13, we can apply the theorem:
\[ 2^{12} \equiv 1 \mod 13 \]

3. **Reduce the Exponent**: We need to reduce \( 5557 \) modulo \( 12 \) (since \( 12 = 13 - 1 \)):
\[ 5557 \div 12 = 463 \quad \text{(quotient)} \quad \text{and} \quad 5557 - (12 \times 463) = 1 \quad \text{(remainder)} \]
Thus, \( 5557 \mod 12 = 1 \).

4. **Calculate the Reduced Power**: Now we can simplify:
\[ 2^{5557} \equiv 2^1 \mod 13 \]

5. **Final Calculation**:
\[ 2^1 = 2 \]
Therefore, the remainder \( R \) when \( 2^{5557} \) is divided by \( 13 \) is:
\[ R = 2 \]

### Conclusion:
The remainder \( R \) when \( \frac{2^{5557}}{13} \) is \( 2 \).