Single correct option
What is the remainder when 11²¹⁰ is divided by 13?
- A. 1
- B. 9
- C. 12
- D. 4
Show answer and solution
Answer: (C) 12
By Fermat's little theorem, 11¹² ≡ 1 (mod 13) since 13 is prime.
210 = 12 × 17 + 6, so 11²¹⁰ ≡ 11⁶ (mod 13).
11² = 121 ≡ 4 (mod 13). 11⁴ ≡ 4² = 16 ≡ 3 (mod 13). 11⁶ = 11⁴ × 11² ≡ 3 × 4 = 12 (mod 13).
Remainder = 12.
210 = 12 × 17 + 6, so 11²¹⁰ ≡ 11⁶ (mod 13).
11² = 121 ≡ 4 (mod 13). 11⁴ ≡ 4² = 16 ≡ 3 (mod 13). 11⁶ = 11⁴ × 11² ≡ 3 × 4 = 12 (mod 13).
Remainder = 12.
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