| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Fermat's Little Theorem |
| Difficulty | Standard +0.3 Part (i) is a direct application of Fermat's Little Theorem requiring only modular arithmetic manipulation (reducing the exponent mod 12, then computing 6^2 mod 13). Part (ii) consists of standard permutation counting problems: (a) is trivial factorial, (b-d) are textbook arrangements with restrictions. All parts require routine application of learned techniques with no novel problem-solving or insight needed, making this slightly easier than average overall. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities7.01d Multiplicative principle: arrangements of n distinct objects7.01g Arrangements in a line: with repetition and restriction8.02l Fermat's little theorem: both forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6^{13-1}\equiv 1\pmod{13}\) or \(6^{13}\equiv 6\pmod{13}\) | B1 | Recalls Fermat's Little Theorem correctly |
| Attempts \(542 = 45\times12+2\) or \(542=41\times13+9\) | M1 | Attempt at \(6^{542}\) in form \((6^{12})^{45}\) or similar |
| \(6^{542}=(6^{12})^{45}\times6^2\) or \(6^{542}=(6^{13})^{41}\times6^9\) | A1 | Uses \(a\) and \(b\) to write \(6^{542}\) correctly in terms of \(6^{12}\) or \(6^{13}\) |
| \(\equiv 1\times6^2\equiv...\pmod{13}\) or \(\equiv 6^{41}\times6^9\equiv(6^{13})^3\times6^2\times6^9\equiv6^{13}\times6\equiv6^2\equiv...\pmod{13}\) | M1 | Completes process to find residue |
| \(\equiv 10\pmod{13}\) | A1 | Correct residue; allow if "45" was incorrect so long as remainder was 2 |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(7! = 5040\) | B1 | Accept as \(7!\) for this part but must be evaluated in remaining parts |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4!\times4! = 576\) | M1 | Evidence 4 students treated as one unit among many; score for \(4!\times k\) where \(k\neq1\) |
| A1 | Correct value | |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5!\times2! = 240\) | M1 | Realises other 5 students can sit in any position — evidenced by sight of \(5!\) |
| A1 | Correct value | |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(7!-6!\times2!=3600\) or \(5!\times(2\times5+2\times4+2\times4+4)=3600\) | M1 | Correct strategy, e.g. answer to (ii)(a) minus ways they sit together; or considers positions for each case |
| A1 | Correct value | |
| (2) |
# Question 4:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6^{13-1}\equiv 1\pmod{13}$ or $6^{13}\equiv 6\pmod{13}$ | B1 | Recalls Fermat's Little Theorem correctly |
| Attempts $542 = 45\times12+2$ or $542=41\times13+9$ | M1 | Attempt at $6^{542}$ in form $(6^{12})^{45}$ or similar |
| $6^{542}=(6^{12})^{45}\times6^2$ or $6^{542}=(6^{13})^{41}\times6^9$ | A1 | Uses $a$ and $b$ to write $6^{542}$ correctly in terms of $6^{12}$ or $6^{13}$ |
| $\equiv 1\times6^2\equiv...\pmod{13}$ or $\equiv 6^{41}\times6^9\equiv(6^{13})^3\times6^2\times6^9\equiv6^{13}\times6\equiv6^2\equiv...\pmod{13}$ | M1 | Completes process to find residue |
| $\equiv 10\pmod{13}$ | A1 | Correct residue; allow if "45" was incorrect so long as remainder was 2 |
| | **(5)** | |
## Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $7! = 5040$ | B1 | Accept as $7!$ for this part but must be evaluated in remaining parts |
| | **(1)** | |
## Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4!\times4! = 576$ | M1 | Evidence 4 students treated as one unit among many; score for $4!\times k$ where $k\neq1$ |
| | A1 | Correct value |
| | **(2)** | |
## Part (ii)(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5!\times2! = 240$ | M1 | Realises other 5 students can sit in any position — evidenced by sight of $5!$ |
| | A1 | Correct value |
| | **(2)** | |
## Part (ii)(d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $7!-6!\times2!=3600$ or $5!\times(2\times5+2\times4+2\times4+4)=3600$ | M1 | Correct strategy, e.g. answer to (ii)(a) minus ways they sit together; or considers positions for each case |
| | A1 | Correct value |
| | **(2)** | |
---\begin{enumerate}
\item (i) Use Fermat's Little Theorem to find the least positive residue of $6 ^ { 542 }$ modulo 13\\
(ii) Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if\\
(a) there are no restrictions where they sit in the row,\\
(b) Alan, Enid, Felix and Graham sit together,\\
(c) Brenda sits at one end of the row and Graham sits at the other end of the row,\\
(d) Charles and Devindra do not sit together.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2019 Q4 [12]}}