| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Euclidean algorithm - HCF only |
| Difficulty | Moderate -0.8 Part (i) is a straightforward application of the divisibility rule for 11 (alternating sum of digits). Part (ii) is a direct, mechanical application of the Euclidean algorithm with no complications—just repeated division with remainder until reaching the HCF. Both parts require only procedural recall with minimal problem-solving, making this easier than average even for Further Maths. |
| Spec | 8.02b Divisibility tests: standard tests for 2, 3, 4, 5, 8, 9, 118.02d Division algorithm: a = bq + r uniquely8.02i Prime numbers: composites, HCF, coprimality |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2-3+7-3+8=\ldots\) or \(2+7+8-(3+3)=\ldots\) | M1 | Executes the correct process by adding and subtracting alternating digits or equivalent |
| \(= 11\) so \(23\,738\) is divisible by \(11\) | A1 | Completes correctly with a correct conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2322 = 3\times654+360,\quad 654=1\times360+294\) | M1 | Uses the Euclidean algorithm showing two stages (Must be Euclidean algorithm, not e.g. using prime factors) |
| \(360=1\times294+66,\quad 294=4\times66+30\) | ||
| \(66=2\times30+6,\quad 30=5\times6+0\) | A1 | Completes the algorithm correctly |
| So \(\text{HCF}(2322,\,654)=6\) | A1 | All correct and concludes HCF is \(6\) |
# Question 1:
## Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2-3+7-3+8=\ldots$ or $2+7+8-(3+3)=\ldots$ | M1 | Executes the correct process by adding and subtracting alternating digits or equivalent |
| $= 11$ so $23\,738$ is divisible by $11$ | A1 | Completes correctly with a correct conclusion |
**(2 marks)**
## Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2322 = 3\times654+360,\quad 654=1\times360+294$ | M1 | Uses the Euclidean algorithm showing two stages (Must be Euclidean algorithm, not e.g. using prime factors) |
| $360=1\times294+66,\quad 294=4\times66+30$ | | |
| $66=2\times30+6,\quad 30=5\times6+0$ | A1 | Completes the algorithm correctly |
| So $\text{HCF}(2322,\,654)=6$ | A1 | All correct and concludes HCF is $6$ |
**(3 marks)**
---\begin{enumerate}
\item (i) Using a suitable algorithm and without performing any division, determine whether 23738 is divisible by 11\\
(ii) Use the Euclidean algorithm to find the highest common factor of 2322 and 654
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 AS 2018 Q1 [5]}}