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\title{Homework 2: Due Friday January 26th 2018\\
MTH 310 001 W18}
\author{YOUR NAME HERE}
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Solutions must be typed using LaTeX.
Remember to provide justification and write in complete sentences for all responses unless otherwise noted. \\
\begin{enumerate}
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% Question 1
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\item Sec. 1.3 Problem 2:
\begin{enumerate}
\item Verify that $2^5-1$ and $2^7-1$ are prime.
\item Show that $2^11-1$ is not prime.
\end{enumerate}
\begin{proof}
Type your answer here!
\end{proof}
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% Question 2
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\item Sec. 1.3 Problem 6: If $p>5$ is prime and $p$ is divided by $10$, show that the remainder is $1,3,7$ or $9$.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 3
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\item Sec. 1.3 Problem 10: Let $p$ be an integer other than $0$, $1$, or $-1$. Prove that $p$ is prime if and only if for each $a\in \bZ$ either $(a,p)=1$ or $p|a$.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 4
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\item Sec. 1.3 Problem 15: If $p$ is prime and $p|a^n$, is it true that $p^n|a^n$? Justify your answer. [Hint: Corollary 1.6]
\begin{proof}
Type your answer here!
\end{proof}
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% Question 5
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\item Sec 1.3 Problem 16: Prove that $(a,b)=1$ if and only if there are no primes such that $p|a$ and $p|b$.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 6
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\item Sec 1.3 Problem 31:
\begin{enumerate}
\item If $p$ is a positive prime, prove that there are no nonzero integers $a$ and $b$ such that $a^2=pb^2$.
\item If $p$ is a positive prime, prove that $\sqrt{p}$ is irrational. [Hint: Use proof by contradiction. Assume $\sqrt{p}=a/b$ with $a,b\in \bZ$ and use part (a)]
\end{enumerate}
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% Question 7
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\item Sec 1.3 Problem 32: Prove that there are infinitely many primes. [Hint: Use proof by contradiction. Assume there are only finitely many primes $p_1,p_2,\dots ,p_k$ and reach a contradiction by showing that the number $p_1\cdot p_2 \cdot \dots \cdot p_k+1$ is not divisible by any of $p_1,p_2,\dots ,p_k$.]
\end{enumerate}
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