Question

对所有实数 $x$ 和满足 $y\neq 0$ 的实数 $y$ ，余数函数可定义为其中 $\left\lfloor \dfrac{x}{y}\right\rfloor$ 表示小于或等于 $\dfrac{x}{y}$ 的最大整数。 $\text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)$ 的值是多少？ $\text{rem}(x,y)=x-y\left\lfloor \dfrac{x}{y}\right\rfloor,$

The remainder function can be defined for all real numbers $x$ and $y$ with $y\neq 0$ by $\text{rem}(x,y)=x-y\left\lfloor \dfrac{x}{y}\right\rfloor,$ where $\left\lfloor \dfrac{x}{y}\right\rfloor$ denotes the greatest integer less than or equal to $\dfrac{x}{y}.$ What is the value of $\text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)?$

$-\dfrac{3}{8}$

$-\dfrac{1}{40}$

$0$

$\dfrac{3}{8}$

$\dfrac{31}{40}$

答案：B

知识点：自定义运算取整函数

难度评级：1200

解答：

首先，所以 $\left\lfloor -\dfrac{15}{16}\right\rfloor=-1$ 。 $\begin{gathered} \dfrac{x}{y}=\dfrac{3/8}{-2/5}\\ =\dfrac{3}{8}\cdot\left(-\dfrac{5}{2}\right)\\ =-\dfrac{15}{16}, \end{gathered}$

因此 $\begin{gathered} \text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)\\ =\dfrac{3}{8}-\left(-\dfrac{2}{5}\right)(-1)\\ =\dfrac{3}{8}-\dfrac{2}{5}\\ =\dfrac{15-16}{40}\\ =-\dfrac{1}{40}. \end{gathered}$

所以正确答案是 B。

First, $\begin{gathered} \dfrac{x}{y}=\dfrac{3/8}{-2/5}\\ =\dfrac{3}{8}\cdot\left(-\dfrac{5}{2}\right)\\ =-\dfrac{15}{16}, \end{gathered}$ and $\left\lfloor -\dfrac{15}{16}\right\rfloor=-1.$

Therefore $\begin{gathered} \text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)\\ =\dfrac{3}{8}-\left(-\dfrac{2}{5}\right)(-1)\\ =\dfrac{3}{8}-\dfrac{2}{5}\\ =\dfrac{15-16}{40}\\ =-\dfrac{1}{40}. \end{gathered}$

Thus, the correct answer is B.

2016 AMC 12A 第 3 题

答案：B

难度评级：1200

解答：

其他年份的第 3 题