Question
Find the remainder when 2⁵⁷ is divided by 5.
Solution
Powers of 2 modulo 5: 2¹ ≡ 2 (mod 5) 2² ≡ 4 (mod 5) 2³ ≡ 8 ≡ 3 (mod 5) 2⁴ ≡ 16 ≡ 1 (mod 5) Pattern repeats every 4: 2, 4, 3, 1 57 ÷ 4 gives remainder 1 So 2⁵⁷ ≡ 2¹ (mod 5) Remainder = 2
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