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2021 AMC 10A Problem 18

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2021 AMC 10A Problem 18AMC 10 Step by Step
Name:                             Date: September 13, 2026
Functions · difficulty 3 of 5 · about 4 min · position P16-20

Let ff be a function defined on the set of positive rational numbers with the property that f(a⋅b)=f(a)+f(b)f(a\cdot b)=f(a)+f(b) for all positive rational numbers aa and bb . Furthermore, suppose that ff also has the property that f(p)=pf(p)=p for every prime number pp . For which of the following numbers xx is f(x)<0f(x)<0 ?

  1. A)

    1732\frac{17}{32}

  2. B)

    1116\frac{11}{16}

  3. C)

    79\frac{7}{9}

  4. D)

    76\frac{7}{6}

  5. E)

    2511\frac{25}{11}

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