\documentclass[12pt]{article} \usepackage{comment} \begin{document} \centerline{\bf Midterm One, March 9 8:00PM-10:00PM} \newcommand{\implies}{\Rightarrow} \newcommand{\MOD}{{\rm MOD}} \newcommand{\PRIMES}{{\rm PRIMES}} \newcommand{\NSQ}{{\rm NSQ}} \newcommand{\into}{{\rightarrow}} \newcommand{\lf}{\left\lfloor} \newcommand{\rf}{\right\rfloor} \newcommand{\lc}{\left\lceil} \newcommand{\rc}{\right\rceil} \newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\Z}{{\sf Z}} \newcommand{\N}{{\sf N}} \newcommand{\Q}{{\sf Q}} \newcommand{\R}{{\sf R}} \newcommand{\Rpos}{{\sf R}^+} \centerline{\bf WARNING: THIS MID IS THREE PAGES LONG!!!!!!!!!!!!!!!!!} \newif{\ifshowsoln} \showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks. \begin{enumerate} \item (15 points) Let $p$ and $q$ be primes. Let $n=p^2q^3$. Show that, $n^{2/5}\notin\Q$. USE Unique Factorization. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (20 points) \begin{enumerate} \item (7 points-1 point each) Fill in the following: 0) $0^4 \equiv \qquad \pmod 8$. 1) $1^4 \equiv \qquad \pmod 8$. 2) $2^4 \equiv \qquad \pmod 8$. 3) $3^4 \equiv \qquad \pmod 8$. 4) $4^4 \equiv \qquad \pmod 8$. 5) $5^4 \equiv \qquad \pmod 8$. 6) $6^4 \equiv \qquad \pmod 8$. 7) $7^4 \equiv \qquad \pmod 8$. \item (13 points) Show that there exists an infinite number of $n$ such that $n$ cannot be written as the sum of 6 fourth powers. (HINT: Use Part a.) \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (15 points) Find a number $M$ such that the following is true, and prove it. $$(\forall n\ge M)(\exists x,y\in\N)[n=37x+38y].$$ \end{enumerate} \end{document}