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\title{Math 214 -- Foundations of Mathematics\\
Homework 6\\
{\large{\bf Due Oct 19, 2012}}}
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\author{Your name}


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Solve the following problems.  Show all your work.  

\begin{enumerate}
\item Let $a,b,c$ be integers. Prove that if $3| (abc-1)$, then $3| (a-1)$,  $3| (b-1)$, or $3| (c-1)$. 


\item Let $d=gcd(a,b)$. If $a=da'$ and $b=db'$, show that $gcd(a',b')=1$. 


\item Let $a,b\in \Z$, where not both $a$ and $b$ are $0$.  Show that there are infinitely many pairs $x,y$ of integers such that $gcd(a,b)=ax+by$. 

\item Show that $n+1$ and $3n+2$ are coprime. 

\item Prove that $\sqrt[3]{3}$ and $\log_{10} 234$ are irrational numbers. 

\item For integers $a$ and $b$, let $lcm(a,b)$ be the least positive multiplier of $a$ and $b$.  Show that $lcm(a,b)\cdot gcd(a,b)=ab$. (hint: express gcd and lcm in terms of the prime factors of $a$ and $b$. you may need to prove that $\max\{m,n\}+\min\{m,n\}=m+n$ first.)




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