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 $f(x) \lt 0?$

$\dfrac{17}{32}$

$\dfrac{11}{16}$

$\dfrac{7}{9}$

$\dfrac{7}{6}$

$\dfrac{25}{11}$

Solution:

The functional equation makes $f$ completely additive: for $x = \prod p^{e_p},$ we have $f(x) = \sum e_p\, f(p) = \sum e_p\, p,$ where a prime in the denominator contributes a negative exponent (since $f(1/p) = -p$ ).

Evaluating: $f\!\left(\tfrac{17}{32}\right) = 17 - 5\cdot 2 = 7,$ $f\!\left(\tfrac{11}{16}\right) = 11 - 4\cdot 2 = 3,$ $f\!\left(\tfrac{7}{9}\right) = 7 - 2\cdot 3 = 1,$ $f\!\left(\tfrac{7}{6}\right) = 7 - 2 - 3 = 2,$ and $f\!\left(\tfrac{25}{11}\right) = 2\cdot 5 - 11 = -1.$ Only the last is negative.

Thus, the correct answer is E.

2021 AMC 12A Spring Problem 18

Solution:

Problem 18 in Other Years