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\noindent \textbf{Math 465 - Spring 2021 \\
 Homework 4} \\
Due February 25th 2021 \vspace{-4mm}\\

\noindent \small \textit{There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?}\\\vspace{1mm} \hfill---
 Sunzi Suanjing, 3rd century AD


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\item Let $m, n > 1$ be (not necessarily coprime) integers. Prove that the two congruences
$x \equiv a~\pmod{m}$ and $x \equiv b \pmod{ n}$ admit a simultaneous solution if and only if $\gcd(m, n)$ divides $a - b$.

\item Use the previous problem to show that there is no simultaneous solution to $x \equiv 5 \pmod{6}$ and $x \equiv 7 \pmod{15}$.

\item Use both Fermat's Little Theorem and the Chinese remainder theorem to show that if $p$ and $q$ are distinct primes then $$ p^{q-1} + q^{p-1} \equiv 1 \pmod{pq}.$$

\item \noindent 5-3.1(a,c) 5-3.4, 5-4.1 (From the textbook).

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