| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2017 |
| Session | Specimen |
| Marks | 14 |
| Topic | Number Theory |
| Type | Modular arithmetic properties |
| Difficulty | Hard +2.3 This is a Further Maths modular arithmetic question requiring multiple proof techniques including Fermat's Little Theorem, systematic case analysis, and combining congruence results. Part (i) requires understanding prime factorization modulo 6, part (ii) needs factorization insight, part (iii) applies FLT with careful exponent manipulation, and part (iv) demands sophisticated use of FLT with Chinese Remainder Theorem reasoning across two moduli—significantly beyond standard A-level. |
| Spec | 8.02e Finite (modular) arithmetic: integers modulo n8.02l Fermat's little theorem: both forms |
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Prove that $p \equiv \pm 1 \pmod{6}$ for all primes $p > 3$. [2]
\item Hence or otherwise prove that $p^2 - 1 \equiv 0 \pmod{24}$ for all primes $p > 3$. [3]
\end{enumerate}
\item Given that $p$ is an odd prime, determine the residue of $2^{p^2-1}$ modulo $p$. [4]
\item Let $p$ and $q$ be distinct primes greater than 3. Prove that $p^{q-1} + q^{p-1} \equiv 1 \pmod{pq}$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2017 Q9 [14]}}