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\begin{document}

\title[101 Homework \#25]{Math 101: Algebra I \\ Homework, Day \#25}
\date{Assigned Friday, 3 November 2017; due Monday, 6 November 2017}

\maketitle

\begin{problab}{25.1}
Let $M$ be the $\Z$-module generated by $x_1,x_2,x_3,x_4$ subject to the relations
\begin{align*} 
x_1+3x_2-9x_3 \phantom{+ 100x_4} &= 0 \\
x_1+3x_2+3x_3 + 12x_4 &= 0 \\
2x_1 + 4x_2 + 2x_3 + 24x_4 &= 0
\end{align*}
Give an explicit isomorphism of $M$ to a direct sum of cyclic abelian groups.  What are the invariant factors and elementary divisors of $\mathrm{Tor}(M)$?  
\end{problab}

\begin{problab}{25.2}
Let $R$ be a PID.  Let $a,b \in R$, not both zero.  Write $(a,b)=(g)$ for $g \in R$, so that there exist $x,y \in R$ such that $ax+by=g$.  Show that $(x,y)=R$.
\end{problab}

\end{document}