# 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)

~Education, the Study of Everything

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