Question

The remainder when \( 64^{64} \) is divided by 7 is equal to:

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

To simplify large powers in modular arithmetic, reduce the base modulo the divisor first, and then raise it to the desired power.

Answer 1

The Correct Option is C

We are asked to find the remainder when \( 64^{64} \) is divided by 7. First, we reduce \( 64 \) modulo 7: \[ 64 \div 7 = 9 \, \text{(quotient)}, \, \text{remainder} = 64 - 9 \times 7 = 64 - 63 = 1 \] Thus, \( 64 \equiv 1 \, (\text{mod} \, 7) \). Now, since \( 64^{64} \equiv 1^{64} \, (\text{mod} \, 7) \), we get: \[ 64^{64} \equiv 1 \, (\text{mod} \, 7) \] Hence, the remainder when \( 64^{64} \) is divided by 7 is \( \boxed{1} \). 
Therefore, the correct answer is (D) 1.