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\begin{document}

\title[Math 250 Handout 2 - Divisibility]{Math 250 Handout 2 - Divisibility}\maketitle

\begin{enumerate}

\item Show that if $d \neq 0$ and $d\mid a$, then $d\mid(-a)$ and $-d\mid a$.

\item Show that if $a\mid b$ and $b\mid a$, then $a=b$ or $a=-b$.

\item Suppose that $n$ is an integer such that $5\mid (n + 2)$. Which of the following are divisible by 5? 
 \begin{enumerate}
 \item [(a)] $n^2 -4$
 \item [(b)] $n^2 +8n+7$
 \item [(c)] $n^4 -1$
 \item [(d)] $n^2 - 2n$ \end{enumerate} 
\item Prove that the square of any integer of the form $5k + 1$ for $k \in \mathbf{Z}$ is of the form $5k' + 1$ for some $k' \in \mathbf{Z}$.

\item Show that if $ ac\mid bc$ and $c \neq 0$, then $a\mid b$.

\vspace{15pt}

\item 
  \begin{enumerate}
 \item [(a)] Prove that the product of three consecutive integers is divisible by 6.
 \item [(b)] Prove that the product of four consecutive integers is divisible by 24.
 \item [(c)] Prove that the product of $n$ consecutive integers is divisible by $n(n-1)$.
 \item [(d)] (Challenge problem) Prove that the product of $n$ consecutive integers is divisible by $n!$.
 
 
 \end{enumerate}

\item Find all integers $n \geq$ 1 so that $n^3 -1$ is prime. Hint: $n^3 -1 = (n^2 +n+1)(n-1)$. 

\item Show that for all integers $a$ and $b$, 
\[
a^2b^2(a^2 - b^2)
\]
is divisible by 12.

% \item Show that for all integers $a$ and $b$, 
% \[
% ab(a^2 - b^2)(a^2 + b^2)
% \]
% is divisible by 30.

\item Suppose that $a$ is an integer greater than 1 and that $n$ is a positive integer. Prove that if $a^n +1$ is prime, then $a$ is even and $n$ is a power of 2. Primes of the form $2^{2^k} + 1$ are called Fermat primes.

\item Suppose that a is an integer greater than 1 and that n is a positive integer. Prove that if $a^n -1$ is prime other than 2, then $a = 2$ and $n$ is a prime. Primes of the form $2^n - 1$ are called Mersenne primes.

\item Let $n$ be an integer greater than 1. Prove that if one of the numbers $2^n - 1, 2^n + 1$ is prime, then the other is composite.

\item Show that every integer of the form $4\cdot 14^k+1$, $k\geq1$ is composite. Hint: show that there is a factor of 3 when $k$ is odd and a factor of 5 when $k$ is even.

\vspace{15pt}

\item Can you find an integer $n > 1$ such that the sum
$$1 + \frac{1}{2}+ \frac{1}{3}+ + \cdots + \frac{1}{n}$$
 is an integer?

\end{enumerate}
\end{document}

