Paper 4, Section II, E

(i) Let $p$ be a prime number, and let $x$ and $y$ be integers such that $p$ divides $x y$. Show that at least one of $x$ and $y$ is divisible by $p$. Explain how this enables one to prove the Fundamental Theorem of Arithmetic.

[Standard properties of highest common factors may be assumed without proof.]

(ii) State and prove the Fermat-Euler Theorem.

Let $1 / 359$ have decimal expansion $0 \cdot a_{1} a_{2} \ldots$ with $a_{n} \in\{0,1, \ldots, 9\}$. Use the fact that $60^{2} \equiv 10(\bmod 359)$ to show that, for every $n, a_{n}=a_{n+179}$.

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