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

To determine the remainder of \( 64^{64} \) divided by 7, we first find the remainder of 64 when divided by 7. \( 64 = 9 \times 7 + 1 \), so \( 64 \equiv 1 \, (\text{mod} \, 7) \). Consequently, \( 64^{64} \equiv 1^{64} \, (\text{mod} \, 7) \), which simplifies to \( 64^{64} \equiv 1 \, (\text{mod} \, 7) \). Thus, the remainder when \( 64^{64} \) is divided by 7 is \( \boxed{1} \).
The correct answer is (D) 1.