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 $f(x) < 0?$

$\dfrac{17}{32}$

$\dfrac{11}{16}$

$\dfrac{7}{9}$

$\dfrac{7}{6}$

$\dfrac{25}{11}$

Answer: E

Concepts:functional equation prime factorization

Difficulty rating: 1280

Video solution:

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Written solution:

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 $f(a/b)=f(a)-f(b).$

Evaluating the choices by prime factorization, $\begin{aligned} f(17/32)&=17-5\cdot2=7,\\ f(11/16)&=11-4\cdot2=3,\\ f(7/9)&=7-2\cdot3=1,\\ f(7/6)&=7-2-3=2,\\ f(25/11)&=2\cdot5-11=-1. \end{aligned}$ Only the final value is negative.

Thus, E is the correct answer.

2021 AMC 10A Spring Problem 18

Answer: E

Difficulty rating: 1280

Video solution:

Written solution:

Problem 18 in Other Years