| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Divisibility proof for all integers |
| Difficulty | Standard +0.3 This is a standard divisibility proof requiring factorisation of a cubic expression and showing divisibility by 2 and 3. Part (a) is straightforward factorisation, and part (b) requires recognising that consecutive integers guarantee factors of 2 and 3, which is a common A-level proof technique but slightly above routine recall. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks | Guidance |
|---|---|---|
| \(n^3 - n = n(n^2-1) = n(n-1)(n+1)\) | B1 | States correct factorisation |
| Answer | Marks | Guidance |
|---|---|---|
| \((n-1)\), \(n\), \((n+1)\) are 3 consecutive integers | E1 | States three consecutive integers; or all integers are multiple of 3 or 1 more/less |
| So one must be a multiple of 3, and at least one must be a multiple of 2 | E1 | Deduces one factor multiple of 3 and one multiple of 2 |
| So product has factors 2 and 3, hence is a multiple of \(2 \times 3 = 6\) | R1 | States at least one multiple of 2, one of 3, draws correct conclusion |
## Question 9(a):
$n^3 - n = n(n^2-1) = n(n-1)(n+1)$ | B1 | States correct factorisation |
## Question 9(b):
$(n-1)$, $n$, $(n+1)$ are 3 consecutive integers | E1 | States three consecutive integers; or all integers are multiple of 3 or 1 more/less |
So one must be a multiple of 3, and at least one must be a multiple of 2 | E1 | Deduces one factor multiple of 3 and one multiple of 2 |
So product has factors 2 and 3, hence is a multiple of $2 \times 3 = 6$ | R1 | States at least one multiple of 2, one of 3, draws correct conclusion |9
\begin{enumerate}[label=(\alph*)]
\item Express $n ^ { 3 } - n$ as a product of three factors.
9
\item Given that $n$ is a positive integer, prove that $n ^ { 3 } - n$ is a multiple of 6
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2021 Q9 [4]}}