Question
What is the remainder when 32025 is divided
by 7?Solution
ATQ, Powers of 3 modulo 7: 3¹ ≡ 3 3² ≡ 9 ≡ 2 3³ ≡ 3·2 = 6 3⁴ ≡ 3·6 = 18 ≡ 4 3⁵ ≡ 3·4 = 12 ≡ 5 3⁶ ≡ 3·5 = 15 ≡ 1 (mod 7) So, 3⁶ ≡ 1 (mod 7), cycle length 6. 2025 ÷ 6 = 337 remainder 3. So 3^2025 ≡ 3³ (mod 7) ≡ 27 ≡ 6 (mod 7).
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