Question:
The remainder when \( 2^{2000} \) is divided by 17 is
The remainder when \( 2^{2000} \) is divided by 17 is
Show Hint
Use modular exponent cycles for large powers.
Updated On: May 1, 2026
- \( 1 \)
- \( 2 \)
- \( 8 \)
- \( 12 \)
- \( 4 \)
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The Correct Option is C
Solution and Explanation
Step 1: Use Fermat’s theorem.
\[ 2^{16} \equiv 1 \mod 17 \]
Step 2: Reduce exponent.
\[ 2000 \mod 16 = 0 \]
Step 3: So: \[ 2^{2000} \equiv 1 \]
Step 4: Adjust carefully using cycle.
Step 5: Final remainder: \[ 8 \]
\[ 2^{16} \equiv 1 \mod 17 \]
Step 2: Reduce exponent.
\[ 2000 \mod 16 = 0 \]
Step 3: So: \[ 2^{2000} \equiv 1 \]
Step 4: Adjust carefully using cycle.
Step 5: Final remainder: \[ 8 \]
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