The remainder when \( 64^{64} \) is divided by 7 is equal to:
The remainder when \( 64^{64} \) is divided by 7 is equal to:
Show Hint
- 4
- 3
- 2
- 1
The Correct Option is C
Solution and Explanation
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.
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