Question

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

• A. \( 4 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 6 \)

Hint:

When working with large exponents modulo a number, simplify the base first and apply properties of exponents to reduce the problem to a manageable level.

Answer 1

The Correct Option is B

The objective is to determine the remainder of \( \left( (64)^{64} \right)^{64} \) when divided by 7.First, we simplify \( 64 \mod 7 \). As \( 64 = 7 \times 9 + 1 \), the remainder is 1:\[64 \equiv 1 \mod 7\]Consequently, \( 64^{64} \equiv 1^{64} \equiv 1 \mod 7 \). By the same logic:\[(64^{64})^{64} \equiv 1^{64} \equiv 1 \mod 7\]
Therefore, the remainder when \( \left( (64)^{64} \right)^{64} \) is divided by 7 is \( 1 \).