| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Moderate -0.8 Part (i) requires only finding a single counterexample (e.g., n=2 gives 11, n=3 gives 29, n=4 gives 83, but n=5 gives 245=5×49), which is straightforward trial. Part (ii) is a simple proof by induction or modular arithmetic showing 3^n cycles through {3,9,7,1} mod 10, never reaching 5. Both parts are routine with minimal steps, significantly easier than typical C3 multi-step calculus problems. |
| Spec | 1.01c Disproof by counter example1.01d Proof by contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| \(3^5 + 2 = 245\) [which is not prime] | M1, A1 [2] | Attempt to find counter-example; correct counter-example identified; If A0, allow M1 for \(3^n + 2\) correctly evaluated for 3 values of \(n\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((3^0 = 1), 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, \ldots\) | M1, A1, B1, B1 [2] | Evaluate \(3^n\) for \(n = 0\) to 4 or 1 to 5; allow just final digit written; \(3^n\) is not divisible by 5; all numbers ending in '5' are divisible by 5. so its last digit cannot be a '5'; must state conclusion for B2 |
| so units digits cycle through \(1, 3, 9, 7, 1, 3, \ldots\) | ||
| so cannot be a '5'. | ||
| OR | ||
| \(3^n\) is not divisible by 5 | ||
| all numbers ending in '5' are divisible by 5. | ||
| so its last digit cannot be a '5' |
### (i)
$3^5 + 2 = 245$ [which is not prime] | M1, A1 [2] | Attempt to find counter-example; correct counter-example identified; If A0, allow M1 for $3^n + 2$ correctly evaluated for 3 values of $n$
### (ii)
$(3^0 = 1), 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, \ldots$ | M1, A1, B1, B1 [2] | Evaluate $3^n$ for $n = 0$ to 4 or 1 to 5; allow just final digit written; $3^n$ is not divisible by 5; all numbers ending in '5' are divisible by 5. so its last digit cannot be a '5'; must state conclusion for B2
so units digits cycle through $1, 3, 9, 7, 1, 3, \ldots$ | |
so cannot be a '5'. | |
**OR** | |
$3^n$ is not divisible by 5 | |
all numbers ending in '5' are divisible by 5. | |
so its last digit cannot be a '5' | |
---\begin{enumerate}[label=(\roman*)]
\item Disprove the following statement:
$3^n + 2$ is prime for all integers $n \geq 0$. [2]
\item Prove that no number of the form $3^n$ (where $n$ is a positive integer) has 5 as its final digit. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2013 Q7 [4]}}