| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Coprimality proofs |
| Difficulty | Challenging +1.2 This is a Further Maths number theory question requiring the Euclidean algorithm and systematic case analysis. While it involves proof techniques beyond standard A-level, the problem is relatively straightforward: apply the algorithm to find hcf(7n+4, 8n+5) = hcf(n+1, 7n+4), then determine when this equals 1. The multi-step reasoning and modular arithmetic place it above average difficulty, but it follows a standard template for coprimality proofs in Further Maths. |
| Spec | 8.02i Prime numbers: composites, HCF, coprimality |
2 Consider the integers $a$ and $b$, where, for each integer $n , \mathrm { a } = 7 \mathrm { n } + 4$ and $\mathrm { b } = 8 \mathrm { n } + 5$. Let $\mathrm { h } = \mathrm { hcf } ( \mathrm { a } , \mathrm { b } )$.
\begin{enumerate}[label=(\alph*)]
\item Determine all possible values of $h$.
\item Find all values of $n$ for which $a$ and $b$ are not co-prime.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2022 Q2 [5]}}