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    OlympiadHardNumber Theory10–12

    How many integers from 1 to 1000 are coprime to 1000?

    Quantos inteiros de 1 a 1000 são coprimos com 1000?

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    Solution

    φ(1000)=1000×(1−12)×(1−15)=1000×12×45=400\varphi(1000) = 1000 \times \left(1-\tfrac{1}{2}\right) \times \left(1-\tfrac{1}{5}\right) = 1000 \times \tfrac{1}{2} \times \tfrac{4}{5} = 400.