| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2020 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Number Theory |
| Type | Divisibility tests and proofs |
| Difficulty | Moderate -0.5 This question tests standard divisibility rules and basic logical deduction. Part (a) requires recognizing that 26 and 13 share factors or using the alternating sum divisibility test for 13. Part (b) applies routine divisibility tests for 4 and 9. Part (c) is straightforward deduction using the given factorization. While it requires knowing specific divisibility tests, the question is largely procedural with no novel problem-solving required, making it slightly easier than average. |
| Spec | 8.02b Divisibility tests: standard tests for 2, 3, 4, 5, 8, 9, 11 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | (a) | 13 divides each pair of digits of N (26, 13, 26, 52) |
| [1] | 2.4 | Or applying a standard divisibility test |
| (b) | 4 | 52 (the final two digits of N) ⇒ 4 |
| 9 | digit-sum of N (= 27) ⇒ 9 | N |
| Since hcf(4, 9) = 1, 4 × 9 = 36 | N | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.4 | Applying these two divisibility tests |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | By Euclid’s Lemma, | |
| 13 | 36 × 725907 and hcf(13, 36) = 1 | |
| ⇒ 13 | 725907 | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.4 |
| 2.2a | M for stating “Euclid’s Lemma” (or full description of its result) |
Question 3:
3 | (a) | 13 divides each pair of digits of N (26, 13, 26, 52) | B1
[1] | 2.4 | Or applying a standard divisibility test
(b) | 4 | 52 (the final two digits of N) ⇒ 4 | N
9 | digit-sum of N (= 27) ⇒ 9 | N
Since hcf(4, 9) = 1, 4 × 9 = 36 | N | B1
B1
B1
[3] | 1.1
1.1
2.4 | Applying these two divisibility tests
Must explain that 4, 9 are co-prime as well as state the
conclusion
(c) | By Euclid’s Lemma,
13 | 36 × 725907 and hcf(13, 36) = 1
⇒ 13 | 725907 | M1
A1
[2] | 2.4
2.2a | M for stating “Euclid’s Lemma” (or full description of its result)
Clear outline of necessary conditions3 In this question, $N$ is the number 26132652.
\begin{enumerate}[label=(\alph*)]
\item Without dividing $N$ by 13, explain why 13 is a factor of $N$.
\item Use standard divisibility tests to show that 36 is a factor of $N$.
It is given that $N = 36 \times 725907$.
\item Use the results of parts (a) and (b) to deduce that 13 is a factor of 725907.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2020 Q3 [6]}}