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{\bf EXAM 1 MATH 431}
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\noindent
{\bf Problem 1.}\;\;In the sudy of Pythagorean numbers the
problem was reduced to finding rational points (i.e.,
points withrational coordinates) in the unit circle $x^2 +
y^2 = 1$. Use the same method to find all rational points
on the curve $x^2 - y^2 = 1$.

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\noindent
{\bf Problem 2.}\;\;Let $p$ be a prime, show that the
simultaneous equations:
\begin{align*}
ax + by &= A\;\; ({\rm mod}\; p)\\
cx + dy &= B\;\; ({\rm mod}\; p)
\end{align*}
has a unique solution modulo $p$ if $ad - bc \ne 0$ (mod
$p$).

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\noindent
{\bf Problem 3.}\;\;Let $p$ be a prime, solve
$$
\sum_{i=1}^{p-1} i^3 = ?\;\; ({\rm mod}\; p).
$$

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\noindent
{\bf Problem 4.}\;\;Find the sum
$$
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + ... +
\frac{1}{(n-1) \cdot n} = ?.
$$
Prove your result.

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\noindent
{\bf Problem 5.}\;\;Three farmers divide equally the rice
that they have grown. One goes to a market where an 83-pound
weight is used, another goes a market where an 110-pound
weight and the last goes to a market where an 135-pound
weight is used. Each farmer sells as many full measure as
possible and at the close of the market the first farmer has
32 pounds of rice left, the second has 70 pounds, and the
third 30 pounds. Find the total amount of rice they took to
the market.

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\noindent
{\bf Problem 6.}\;\;Show that if $p$ is prime then for any
integers $a$ and $b$, $(a + b)^p = a + b$ (mod $p$).


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