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\title{CSCI 301 - Assignment 4, Fall 2024}
\author{Your name here}
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Modify the .tex source file for this document, replacing the placeholders with
your solutions. \\
This is an individual assignment. See the syllabus for the collaboration policy.\\

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\begin{enumerate}
\item (5 points) Let $a, b \in \mathbb{R}$ and $a \ne 0$. Prove that if $a$ is rational and $ab$ is irrational, then $b$ is irrational.

\textbf{Proof:} Replace this text with your proof.

\item (5 points) Prove or disprove the following: \\
\textbf{Proposition:} There exist two prime numbers $p$ and $q$ for which $p - q = 97$.

This statement is \textit{(replace this text with true or false)}. 

\textbf{Proof:} Replace this text with your proof.

\item (5 points) Prove that $|P(A)| = 2^{|A|}$. \textit{Hint:} Think back to Lab 2.

\textbf{Proof:} Replace this text with your proof.

\item (5 points) \textbf{Definition:} A number $n$ is \textbf{prime} if it has exactly two positive divisors, $1$ and $n$. If it has more than two positive divisors, it is \textbf{composite}.

Here's an important theorem you may have seen before:

\textbf{Fundamental Theorem of Arithmetic:} Any integer greater than 1 has a \textbf{unique} prime factorization.

Prove the following \textit{part} of the Fundamental Theorem of Arithmetic: Any integer $n$ greater than 1 has \textbf{a} prime factorization (you need not prove its uniqueness). In other words, $n$ can be written as a product of prime factors, as in $n = p_1 p_2 \cdots p_k$, where $p_i$ is prime for all $1 \le i \le k$.

\textbf{Proof:} Replace this text with your proof.

\end{enumerate}

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