2021 AMC 10A Spring Problem 18

Question

Let

f

be a function defined on the set of positive rational numbers with the property that

f(a\cdot b)=f(a)+f(b)

for all positive rational numbers

a

and

b.

Suppose that

f

also has the property that

f(p)=p

for every prime number

p.

For which of the following numbers

x

is

f(x) < 0?

\dfrac{17}{32}

\dfrac{11}{16}

\dfrac{7}{9}

\dfrac{7}{6}

\dfrac{25}{11}

First determine

f(p^e)

and

f(\frac{a}{b})

Evaluate the choices by prime factorization

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Repeated use of the functional equation gives

f(p^e)=ef(p)=ep

for every prime

p

and positive integer

e.

Also,

f(a)=f\left(\frac ab\right)+f(b),

so

f(\frac{a}{b})=f(a)-f(b).

Evaluating the choices by prime factorization,

\begin{aligned} f(\frac{17}{32})&=17-5\cdot2=7,\\ f(\frac{11}{16})&=11-4\cdot2=3,\\ f(\frac{7}{9})&=7-2\cdot3=1,\\ f(\frac{7}{6})&=7-2-3=2,\\ f(\frac{25}{11})&=2\cdot5-11=-1. \end{aligned}

Only the final value is negative.

Thus, E is the correct answer.

Problem 18 in other years