Question

设 $f$ 是一个函数，对所有正实数 $x$ 和 $y$ 都满足 $f(xy) = \dfrac{f(x)}{y}$ 。若 $f(500) = 3$ ，则 $f(600)$ 的值是多少？

Let $f$ be a function satisfying $f(xy) = \dfrac{f(x)}{y}$ for all positive real numbers $x$ and $y.$ If $f(500) = 3,$ what is the value of $f(600)?$

$1$

$2$

$\dfrac{5}{2}$

$3$

$\dfrac{18}{5}$

取 $x = 500$ ，且令 $y = \dfrac{6}{5}$ ，使得 $xy = 600$ 。则 $f(600) = \dfrac{f(500)}{6/5} = \dfrac{3}{6/5} = \dfrac{5}{2}.$

因此，正确答案是 C。

Choose $x = 500$ and $y = \dfrac{6}{5}$ so that $xy = 600.$ Then $f(600) = \dfrac{f(500)}{6/5} = \dfrac{3}{6/5} = \dfrac{5}{2}.$

Thus, the correct answer is C.