ProblemSet3b.tex

\documentclass[letterpaper,12pt]{article}
\usepackage{amsmath, amssymb}

\oddsidemargin=-.25in 
\evensidemargin=-.25in 
\topmargin=-.75in 
\textwidth=7in 
\textheight=10in

\def\p{\varphi}
\def\s{\sigma}
\def\R{\mathbb R}
\def\N{\mathbb N}
\def\Q{\mathbb Q}
\def\Z{\mathbb Z}
\def\C{\mathbb C}





\begin{document}
\begin{enumerate}
\item
Let $a$ be a square-free positive integer (square-free means for all $k\in \Z$ if $k^2|a$, then $k=\pm 1$). Let $p_1,\dots,p_n$ be primes such that $a=p_1p_2\cdots p_n$. Prove that the number of divisors of $a$ is $2^n$.

\item
Let $a$ be a positive integer, and let $p_1,\dots,p_n$ be distinct primes such that $a=p_1^{a_1}p_2^{a_2}\cdots p_n^{a_n}$. Prove that the number of divisors of $a$ is $\prod_{i=1}^n(a_i+1)$.

\end{enumerate}
\end{document}