2016 AMC 12A Problems/Problem 3

Problem

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

$\textbf{(A)}\ -\frac{3}{8}\qquad\textbf{(B)}\ -\frac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \frac{3}{8}\qquad\textbf{(E)}\ \frac{31}{40}$

Solution

$\text{rem}\left(\frac{3}{8},-\frac{2}{5}\right)$ $=\frac{3}{8}-\left(-\frac{2}{5}\right)\Big\lfloor\frac{\frac{3}{8}}{-\frac{2}{5}}\Big\rfloor$ $=\frac{3}{8}+\left(\frac{2}{5}\right)\Big\lfloor -\frac{15}{16}\Big\rfloor$ $=\frac{3}{8}+\left(\frac{2}{5}\right)\left(-1\right)$ $=\frac{3}{8}-\frac{2}{5}$ $=\boxed{\textbf{(B)}-\frac{1}{40}}$

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2016 AMC 12A Problems/Problem 3

Problem

Solution