Find the remainder when `(2)^(5555)` is divided by 13.

Options

A. 4
B. 5
C. 6
D. 7

Answer

To find the remainder when \(2^{5555}\) is divided by 13, 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, \(a = 2\) and \(p = 13\). Since 2 is not divisible by 13, we can apply the theorem.

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

\[ 2^{12} \equiv 1 \mod 13 \]

### Step 2: Reduce the exponent modulo 12
Next, we need to reduce the exponent \(5555\) modulo \(12\) because \(2^{12} \equiv 1\):

\[ 5555 \mod 12 \]

To find \(5555 \mod 12\), we can perform the division:

\[ 5555 \div 12 = 462.9167 \quad \text{(take the integer part, which is 462)} \]
\[ 462 \times 12 = 5544 \]
\[ 5555 - 5544 = 11 \]

Thus,

\[ 5555 \mod 12 = 11 \]

### Step 3: Calculate \(2^{11} \mod 13\)
Now we need to calculate \(2^{11} \mod 13\):

\[ 2^{11} = 2048 \]

Now we find \(2048 \mod 13\):

\[ 2048 \div 13 = 157.5385 \quad \text{(take the integer part, which is 157)} \]
\[ 157 \times 13 = 2041 \]
\[ 2048 - 2041 = 7 \]

Thus,

\[ 2^{11} \equiv 7 \mod 13 \]

### Conclusion
The remainder when \(2^{5555}\) is divided by 13 is:

\[ \boxed{7} \]