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{\bf EXAM 2 MATH 431}
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{\bf Problem 1.}\;\;Let $p$ be a prime. (i) Show that $(a+b)^p = a^p +
b^p\;\; ({\rm mod}\; p)$. (ii) Is it possibel that $\binom{p}{k} = 2\;\;
({\rm mod}\; p)$?

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{\bf Problem 2.}\;\;Solve the simultaneous equations:
\begin{align*}
x &= 3\;\;
({\rm mod}\; 37)\\
x &= 1\;\;
({\rm mod}\; 87)
\end{align*}

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{\bf Problem 3.}\;\;Let $a, b, c$ be primitive solutions of the equation
$$
a^2 + b^2 = c^2.
$$
Is it possible that both $a$ and $b$ are even? Is it possible that both
$a$ and $b$ are odd?

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{\bf Problem 4.}\;\;State and prove your favorite Theorem that you leaned
in this course.


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