Question
Find the remainder when 32025 is divided by
35.Solution
We can use Euler’s theorem or cyclicity. φ(35) = φ(5 × 7) = 4 × 6 = 24 So 324 ≡ 1 (mod 35), and powers repeat every 24. Compute exponent modulo 24: 2025 ÷ 24 = 84 × 24 = 2016, remainder 9 So 32025 ≡ 39 (mod 35) Now compute 39 mod 35: 3¹ = 3 3² = 9 3³ = 27 3⁴ = 81 ≡ 81 − 70 = 11 3⁵ = 11×3 = 33 3⁶ = 33×3 = 99 ≡ 99 − 70 = 29 3⁷ = 29×3 = 87 ≡ 87 − 70 = 17 3⁸ = 17×3 = 51 ≡ 51 − 35 = 16 3⁹ = 16×3 = 48 ≡ 48 − 35 = 13 Remainder = 13.
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