Question
Find the remainder when 3¹²⁵ is divided by 13.
Solution
φ(13) = 12 (since 13 is prime). By Euler’s theorem: 3¹² ≡ 1 (mod 13). Now 125 = 12×10 + 5 So 3¹²⁵ = (3¹²)¹⁰ × 3⁵ ≡ 1¹⁰ × 3⁵ (mod 13) Compute 3⁵: 3² = 9 3³ = 27 ≡ 1 (mod 13) So 3⁵ = 3² × 3³ ≡ 9 × 1 = 9 (mod 13)
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