What is the remainder when 2^{1000000} is divided by 7?

Explanation

Using cyclicity, 2^3 = 8 \equiv 1 \pmod{7}. Since 1000000 = 3 \times 333333 + 1, we have 2^{1000000} = (2^3)^{333333} \times 2^1 \equiv 1^{333333} \times 2 \equiv 2 \pmod{7}.

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