HW5 Problem Set

\begin{document} \maketitle For your convenience, here is transcript of the problems for this homework. \begin{itemize} \item[6.8] Use Euclid's Algorithm to find the GCD and check by the PFT. \item[6.8a] $gcd(126,224)$ \item[6.8b] $gcd(221,299)$ \item[6.9] Find all solutions for these Diophantine equations. \item[6.9a] $17x+13y=220$ \item[6.9b] $21x+15y=93$ \item[6.9c] $60x+42y=104$ \item[6.9d] $588x+231y=63$ \item[6.17] Prove that $gcd(a+b,a-b)=gcd(2a,a-b)=gcd(a+b, 2b)$ \item[6.18] Suppose $gcd(a,b)=1$. Does this determine $gcd(a^2,b^2)$? $gcd(a,2b)$?. \item[6.28] Suppose that $gcd(a,b)=1 \wedge a|n \wedge b|n \Rightarrow ab|n$. (If you don't use the PFT for this, then you are not following directions. \item[Extra Credit] Why is the first hypothesis necessary? \end{itemize} \end{document}