\documentclass[a4paper]{article}
\usepackage[english]{babel}
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\usepackage{amsmath}
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\title{Assignment 0}
\author{Jacob Vella}
\date{\today}
\begin{document}
\maketitle
\section{Problem 1}
Theorem:
if $n$ is odd than $n^2$ is odd
Direct proof:
if $n$ is odd, then $n=2k+1$
if $n^2$ is odd, then $n^2=2k+1$
$(2k+1)^2 = 2k$
$(2k+1)(2k+1) = 2k+1$
$4k^2+2k+2k+1 = 2k+1$
$2(2k^2+k+k)+1$
$2k+1 = 2k+1$
\section{Problem 2}
Theorem: If $a,b > 225$ and $a,b \in {N}^+$, then either $a > 15$ or $b > 15$
Proof by contraposition: $(a \le 15)$ and $(b \le 15)$ then $(ab \le 225)$
if $a = 15$ and $b = 15$
then $ab \ge 225$
\end{document}
