| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Euclidean algorithm - HCF only |
| Difficulty | Moderate -0.5 Part (i) is a straightforward application of the Euclidean algorithm with small numbers requiring only mechanical division steps. Part (ii) is a standard permutation counting problem with clear constraints. Both parts require only routine application of well-practiced techniques with no novel insight, making this easier than average but not trivial due to the multi-step nature and potential for arithmetic errors. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities7.01e Permutations: ordered subsets of r from n elements8.02d Division algorithm: a = bq + r uniquely |
| Answer | Marks |
|---|---|
| M1 | Attempts Euclid's algorithm (there may be an arithmetic slip finding 119) |
| M1 | Uses Euclid's algorithm a further two times with 161 and "their 119" and then with "their 119" and "their 42" |
| A1 | This should be accurate with all the steps shown |
| Answer | Marks |
|---|---|
| B1 | Correctly interprets the problem and uses the five odd digits and four even digits to form a correct product |
| B1 | Interprets the new situation using the four even digits, then the seven digits which have not been used, to form a correct product |
| M1 | Subtracts one answer from the other |
| A1 | Correct answer |
# Question 1
## 1(i)
M1 | Attempts Euclid's algorithm (there may be an arithmetic slip finding 119)
M1 | Uses Euclid's algorithm a further two times with 161 and "their 119" and then with "their 119" and "their 42"
A1 | This should be accurate with all the steps shown
**(3 marks)**
## 1(ii)
B1 | Correctly interprets the problem and uses the five odd digits and four even digits to form a correct product
B1 | Interprets the new situation using the four even digits, then the seven digits which have not been used, to form a correct product
M1 | Subtracts one answer from the other
A1 | Correct answer
**(4 marks)**
**Total: 7 marks**\begin{enumerate}
\item (i) Use the Euclidean algorithm to find the highest common factor of 602 and 161.
\end{enumerate}
Show each step of the algorithm.\\
(ii) The digits which can be used in a security code are the numbers $1,2,3,4,5,6,7,8$ and 9.
Originally the code used consisted of two distinct odd digits, followed by three distinct even digits.
To enable more codes to be generated, a new system is devised. This uses two distinct even digits, followed by any three other distinct digits. No digits are repeated.
Find the increase in the number of possible codes which results from using the new system.
\hfill \mbox{\textit{Edexcel FP2 Q1 [7]}}