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

We are tasked with finding the remainder when \( \left( (64)^{64} \right)^{64} \) is divided by 7. We begin by reducing \( 64 \mod 7 \). Since \( 64 \div 7 = 9 \) with a remainder of 1, we have: \[ 64 \equiv 1 \mod 7 \] This means that \( 64^{64} \equiv 1^{64} \equiv 1 \mod 7 \), and similarly: \[ (64^{64})^{64} \equiv 1^{64} \equiv 1 \mod 7 \]
Thus, the remainder when \( \left( (64)^{64} \right)^{64} \) is divided by 7 is \( 1 \).

Answer 2

\[ 64^{64} \Rightarrow (63 + 1)^{64} = 63\lambda + 1 \] \[ 64^{64^{64}} \Rightarrow (63 + 1)^{64^{64}} = 63\lambda_1 + 1 \] Therefore, the required remainder when divided by \(7\) is: \[ \boxed{1} \]