Question

El residuo se puede definir para todos los números reales $x$ y $y$ con $y \neq 0$ mediante $\text{rem} (x ,y)=x-y\left \lfloor \dfrac{x}{y} \right \rfloor$ donde $\left \lfloor \frac{x}{y} \right \rfloor$ denota el mayor entero menor o igual que $\frac{x}{y}$ . ¿Cuál es el valor de $\text{rem} \left(\frac{3}{8}, -\frac{2}{5} \right)$ ?

The remainder 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 $\left \lfloor \frac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\frac{x}{y}.$ What is the value of $\text{rem} \left(\frac{3}{8}, -\frac{2}{5} \right)?$

$-\dfrac{3}{8}$

$-\dfrac{1}{40}$

$0$

$\dfrac{3}{8}$

$\dfrac{31}{40}$

Solución:

Usando la fórmula, obtenemos $\begin{align*} \text{rem} \left(\dfrac{3}{8}, -\dfrac{2}{5}\right) &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor \dfrac{\frac{3}{8}}{-\frac{2}{5}}\right \rfloor \\ &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor -\dfrac{15}{16} \right \rfloor \\ &= \dfrac{3}{8} + \dfrac{2}{5} \cdot -1 \\ &= \dfrac{3}{8} - \dfrac{2}{5} \\ &= - \dfrac{1}{40} \end{align*}$ Por lo tanto, la respuesta correcta es B.

Using the formula, we get $\begin{align*} \text{rem} \left(\dfrac{3}{8}, -\dfrac{2}{5}\right) &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor \dfrac{\frac{3}{8}}{-\frac{2}{5}}\right \rfloor \\ &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor -\dfrac{15}{16} \right \rfloor \\ &= \dfrac{3}{8} + \dfrac{2}{5} \cdot -1 \\ &= \dfrac{3}{8} - \dfrac{2}{5} \\ &= - \dfrac{1}{40} \end{align*}$ Thus, the correct answer is B .

2016 AMC 10A Problema 4

Solución:

El Problema 4 en otros años