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{\Huge
15-151 Homework 1
}

Please submit in class at 8:00am on Thursday 6th July
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\subsection*{Exercises}

\begin{enumerate}[1.]
\item Prove that if an integer $a$ leaves a remainder of $r$ when divided by an integer $b$, then it leaves a remainder of $r$ when divided by $-b$.
\points{10}

\item Let $a, b \in \mathbb{Z}$. Prove that if $d$ divides $a$ and $d$ divides $b$, then $d$ divides $au+bv$, where $u$ and $v$ are any integers.
\points{10}
\end{enumerate}

\subsection*{Other tasks}

\begin{enumerate}[1.]
\setcounter{enumi}{2}
\item Read the syllabus and sign the attached agreement form. \points{5}
\item Register for a (free) account at ShareLaTeX (\url{https://www.sharelatex.com/}). Create a new project, and upload the file \texttt{template.tex} (available on the course web page) to the project. Change the name and date to your name and today's date. Press `Recompile' and download a PDF of the template file. Send the resulting PDF file to \texttt{cnewstead@cmu.edu} with your name in the subject line. \points{5}
\item Fill out the questionnaire located at \url{https://goo.gl/forms/zu9n74HL9ucjzYp63} \points{5}
\end{enumerate}

\subsection*{Optional but recommended tasks {\normalfont (not for credit)}}

\begin{enumerate}[1.]
\setcounter{enumi}{5}
\item Go say hi to Alp in Doherty Hall 1217 during academic hours this evening (6:30--9:00pm). Alp a Mathematics major who probably has a lot to say about his experience at CMU, the college admissions process, and so on! He will go over some of the concepts from class during the first ${\sim}45$ minutes, and will be available to answer questions for the rest of the time.
\item In this course, we adopt the convention that $0$ is a natural number. Some people prefer to exclude $0$, so that the natural numbers start at $1$. Think of a reason why someone might want to consider $0$ as a natural number, and a reason why someone might want to exclude it. What problems might this mismatch raise for mathematicians? Does this make mathematics less precise? Why or why not?
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