| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 5 |
| Topic | Proof |
| Type | Divisibility proof for all integers |
| Difficulty | Standard +0.8 This is a two-part proof question requiring algebraic manipulation and number theory insight. Part (a) is straightforward (factor theorem or polynomial division), but part (b) requires recognizing that 2^8 ≡ 1 (mod 17) and applying the result from (a) with substitution x=2^8, n=k. The connection between parts requires mathematical maturity beyond routine exercises, though the techniques themselves are A-level standard. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
\begin{enumerate}[label=(\alph*)]
\item Show that, if $n$ is a positive integer, then $(x^n - 1)$ is divisible by $(x - 1)$. [1]
\item Hence show that, if $k$ is a positive integer, then $2^{8k} - 1$ is divisible by 17. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2018 Q7 [5]}}