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\title{Math 412  Homework 1}
\date{Due date: 11:59pm Sept 2, 2016}
\author{your name}


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\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. Each problem is 4 points.



\begin{enumerate}
\item Show that if $2 | n$ and $3 | n$, then $6 | n$.  

\item Show that the square of every odd integer is of the form $8k+1$.  

\item Show that $11 | a^2+5 b^2$ if and only if $11 | a$ and $11 | b$ for $a,b\in Z$. 


\item Show that $8a+3$ and $5a+2$ are relatively prime for all integers $a$. 

\item Show that $(ac, bc)=|c|(a,b)$ for any integers $a, b, c$. 

\item For the following linear diophantine equation, either find all solutions or show that there is no integral solutions:
$$25x+95y=970$$

\item (Bonus) Suppose that the coefficients of the polynomial $f(x)$ are integers. If $f(a)=f(a+2)=(a+1)^2$ for some positive integer $a$ and $0<f(0)<(a+1)^2$, find the value of $f(0)$. 


  
\end{enumerate}

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