| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Wilson's Theorem |
| Difficulty | Hard +2.3 This is a Further Maths question requiring Wilson's Theorem application and modular arithmetic insight. Part (a) demands a non-trivial proof connecting Wilson's Theorem to the given congruence. Part (b) requires recognizing that testing small primes p involves checking if 2×34^34 ≡ 2^15 (mod p), which needs sophisticated modular arithmetic manipulation. The combination of proof and problem-solving with advanced number theory places this well above typical A-level difficulty. |
| Spec | 8.02l Fermat's little theorem: both forms8.02m Order of a modulo p: p-1 not necessarily least such n |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (b) | Alt. (for factor 19) |
| Answer | Marks | Guidance |
|---|---|---|
| 2×3434 −215 = 235 × 1734 – 215 = 233 2×1717 – 215 | B1 | Indices work to obtain a part (a) expression |
| ≡ 233(–1)2 – 215 (mod 19) | M1 | Using (a)’s result |
| 233 – 215 = 215(218 – 1) | M1 | Working mod 19 with remaining numerical term(s) |
| 218 – 1 ≡ 0 (mod 19) by FLT since hcf(2, 19) = 1 | M1 | Use of FLT or calculator |
| and N is a multiple of 19 | A1 | Valid conclusion (FLT justified or correctly demonstrated numerical |
| Answer | Marks |
|---|---|
| NB N = 215 × 3 × 19 × 389 × … | [7] |
Question 8:
8 | (b) | Alt. (for factor 19)
( )2
2×3434 −215 = 235 × 1734 – 215 = 233 2×1717 – 215 | B1 | Indices work to obtain a part (a) expression
≡ 233(–1)2 – 215 (mod 19) | M1 | Using (a)’s result
233 – 215 = 215(218 – 1) | M1 | Working mod 19 with remaining numerical term(s)
218 – 1 ≡ 0 (mod 19) by FLT since hcf(2, 19) = 1 | M1 | Use of FLT or calculator
and N is a multiple of 19 | A1 | Valid conclusion (FLT justified or correctly demonstrated numerical
work)
Note that 218 – 1 = 262 143 = 19 × 13 797 and
233 – 215 = 8589901824 = 19 × 452100096
NB N = 215 × 3 × 19 × 389 × … | [7]
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© OCR 20198 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Prove that $2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )$, where $p$ is an odd prime.
\item Find two odd prime factors of the number $N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }$.
\section*{END OF QUESTION PAPER}
\section*{OCR \\
Oxford Cambridge and RSA}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2019 Q8 [11]}}