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\noindent{\Large{\textsc{Intro to Abstract Mathematics\hfill Assignment 7.2  \\ Spring 2020 \hfill Due March 18}}}

\large

\vspace{1cm}


\begin{exercise}
Prove that for all $a, b, c \in \Z$, if $c|a$ and $c|b$, then $c|(xa + yb)$ for all $x, y \in \Z$. 
\end{exercise}

\begin{exercise}  
Let $a, b \in \Z$ and suppose that there exist $x, y \in \Z$ so that $xa + yb = 1$.  Prove that 1 is the only common divisor of $a$ and $b$.
\end{exercise}

\begin{exercise}
Let $a, b, c \in \Z$.  Prove that if $a$ divides either $b$ or $c$, then $a$ divides $bc$.  Is the converse true?  Justify your answer.
\end{exercise}

\begin{exercise}
Let $A = \{a \in \N \, | \, a \text{ is odd} \}$, $B = \{ x^2 + y^2 \, | \, x, y \in \N \}$ and $C = \{ 4k+1 \, | \, k \in \N \}$.  Prove that $A \cap B \subset C$.  Are the sets equal?  Justify your answer.  [{\em Suggestion:}  Notice that if $a$ is odd and $a = x^2 + y^2$, then one of $x^2$ and $y^2$ must be even, and the other must be odd.]
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