MathDB

4

Part of 2016 AMC 10

Problems(2)

Remainder equation

Source: Problem #4 2016 10A and 2016 12A #3

2/3/2016
The remainder can be defined for all real numbers xx and yy with y≠0y \neq 0 by rem(x,y)=x−y⌊xy⌋\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor where ⌊xy⌋\left \lfloor \tfrac{x}{y} \right \rfloor denotes the greatest integer less than or equal to xy\tfrac{x}{y}. What is the value of rem(38,−25)\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )?
<spanclass=′latex−bold′>(A)</span>−38<spanclass=′latex−bold′>(B)</span>−140<spanclass=′latex−bold′>(C)</span>0<spanclass=′latex−bold′>(D)</span>38<spanclass=′latex−bold′>(E)</span>3140<span class='latex-bold'>(A) </span> -\frac{3}{8} \qquad <span class='latex-bold'>(B) </span> -\frac{1}{40} \qquad <span class='latex-bold'>(C) </span> 0 \qquad <span class='latex-bold'>(D) </span> \frac{3}{8} \qquad <span class='latex-bold'>(E) </span> \frac{31}{40}
floor function2016 AMC 10A
Zoey The Avid Reader

Source: Probem #4 2016 AMC 10B

2/21/2016
Zoey read 1515 books, one at a time. The first book took her 11 day to read, the second book took her 22 days to read, the third book took her 33 days to read, and so on, with each book taking her 11 more day to read than the previous book. Zoey finished the first book on a monday, and the second on a Wednesday. On what day the week did she finish her 1515th book?
<spanclass=′latex−bold′>(A)</span> Sunday<spanclass=′latex−bold′>(B)</span> Monday<spanclass=′latex−bold′>(C)</span> Wednesday<spanclass=′latex−bold′>(D)</span> Friday<spanclass=′latex−bold′>(E)</span> Saturday<span class='latex-bold'>(A)</span>\ \text{Sunday}\qquad<span class='latex-bold'>(B)</span>\ \text{Monday}\qquad<span class='latex-bold'>(C)</span>\ \text{Wednesday}\qquad<span class='latex-bold'>(D)</span>\ \text{Friday}\qquad<span class='latex-bold'>(E)</span>\ \text{Saturday}
AMCAMC 10AMC 10 B2016 AMC 10B