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\title{\textbf{Homework 2}}

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\section*{Section 2.4}
\textbf{Exercise 2.}\\
Prove that $$1^3+2^3+ \ldots n^3 = \displaystyle \frac{n^2(n+1)^2}{4} $$ for $n \in \N$.
\vskip1ex
\emph{Proof: } \vskip2ex  \hfill $\square$

\vskip2ex
\textbf{Exercise 5.}\\
Prove that $10^{n+1} + 10^n + 1$ is divisible by $3$ for $n \in \N$.
\vskip1ex
\emph{Proof: } \vskip2ex  \hfill $\square$

\vskip2ex
\textbf{Exercise 12.}\\
Let $X$ be a set. For every positive integer $n$, show that a set with exactly $n$ elements has a power set with exactly $2^n$ elements.
\vskip1ex
\emph{Proof: } \vskip2ex  \hfill $\square$

\vskip2ex
\textbf{Exercise 15(d, f).}\\
Calculate gcd$(a,b)$ and find integers $r$ and $s$ such that gcd$(a,b) = ra+sb$ if
\begin{enumerate}
\item[(d)] $a = 471$ and $b = 562$
\item[(f)] $a = -4357$ and $b = 3754$
\end{enumerate}
\vskip1ex
\emph{Solution: } \vskip2ex
\hfill $\square$

%\vskip2ex
%\textbf{Exercise 16.}\\
%Let $a$ and $b$ be nonzero integers. If there exist integers $r$ and $s$ such that $ar+bs = 1$, show that $a$ and $b$ are relatively prime.
%\vskip1ex
%\emph{Proof: } \vskip2ex
%\hfill $\square$

\vskip2ex
\textbf{Exercise 18.}\\
Let $a$ and $b$ be integers such that gcd$(a,b) = 1$. Let $r$ and $s$ be integers such that $ra+sb = 1$. Prove that  gcd$(a,s)=$ gcd$(r,b)=$ gcd$(r,s) = 1.$ 
\vskip1ex
\emph{Proof: } \vskip2ex
\hfill $\square$

%\vskip2ex
%\textbf{Exercise 23.}\\
%Prove that there exists a unique least common multiple for any two integers $a$ and $b$.
%\vskip1ex
%\emph{Proof: }
%\vskip2ex
%\hfill $\square$

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