2016 AMC 10B Problems/Problem 4

Problem

Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ 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 $15$ th book?

$\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Solution 1

The process took $1+2+3+\ldots+13+14+15=120$ days, so the last day was $119$ days after the first day. Since $119$ is divisible by $7$ , both must have been the same day of the week, so the answer is $\boxed{\textbf{(B)}\ \text{Monday}}$

Solution 2

Similar to solution 1, the process took 120 days. $120 \equiv 1 \mod 7$ . Since Zoey finished the first book on Monday and the second book (after three days) on Wednesday, we conclude that the modulus must correspond to the day (e.g., $1\mod 7$ corresponds to Monday, $4\mod 7$ corresponds to Thursday, $0\mod 7$ corresponds to Sunday, etc.). The solution is therefore $\boxed{\textbf{(B)}\ \text{Monday}}$ .

Video Solution (CREATIVE THINKING)

https://youtu.be/XG5fR4xl1o4

~Education, the Study of Everything

Video Solution

https://youtu.be/8_xEaEIJZ24

~savannahsolver

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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