2021 AMC 10A Problem 18

Problem 18 of 25IntermediateAlgebra Number Theory

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

f(x) < 0?

Show solution

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),

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.

Tagged: functional equation · prime factorization