Question
Find the remainder when (2^50 + 2^75) is divided by 7.
Solution
2^1 mod 7 = 2 2^2 mod 7 = 4 2^3 mod 7 = 8 mod 7 = 1 So powers of 2 modulo 7 repeat in cycle (2, 4, 1) of length 3. 50 mod 3 = 2 => 2^50 mod 7 = 2^2 mod 7 = 4 75 mod 3 = 0 => 2^75 mod 7 = 2^3 mod 7 = 1 (2^50 + 2^75) mod 7 = (4 + 1) mod 7 = 5
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