Single correct option
What is the remainder when 2¹⁰⁰ is divided by 7?
- A. 1
- B. 2
- C. 3
- D. 4
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Answer: (B) 2
Powers of 2 modulo 7 repeat in a cycle: 2¹ ≡ 2, 2² ≡ 4, 2³ = 8 ≡ 1.
Since 2³ ≡ 1, write 2¹⁰⁰ = (2³)³³ × 2¹ ≡ 1 × 2 = 2 (mod 7).
Tip: Find the smallest power that leaves remainder 1. Then reduce the exponent modulo that cycle length (100 mod 3 = 1).
Since 2³ ≡ 1, write 2¹⁰⁰ = (2³)³³ × 2¹ ≡ 1 × 2 = 2 (mod 7).
Tip: Find the smallest power that leaves remainder 1. Then reduce the exponent modulo that cycle length (100 mod 3 = 1).
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