Question

The remainder, when \( 2^{100} \) is divided by 11, is:

• A. 2
• B. 1
• C. 3
• D. 5

Hint:

Fermat's Little Theorem is an excellent shortcut for finding remainders of large powers divided by prime numbers.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We can use Fermat's Little Theorem, which states that if \( p \) is a prime number, then for any integer \( a \) not divisible by \( p \), \( a^{p-1} \equiv 1 \pmod{p} \).

Step 2: Key Formula or Approach:
Here, \( a = 2 \), \( p = 11 \). Since 11 is prime, \( 2^{10} \equiv 1 \pmod{11} \).

Step 3: Detailed Explanation:
\[ 2^{100} = (2^{10})^{10} \]
Using the theorem: \( (2^{10})^{10} \equiv 1^{10} \pmod{11} \).
\[ 1^{10} = 1 \]
Therefore, the remainder is 1.

Step 4: Final Answer:
The remainder is 1.