AS June 2019 Q2
2.
(i) Determine all the possible integers \(a\), where \(a \gt 3\), such that\[15 \equiv 3 \bmod a\] (2)
(ii) Show that if \(p\) is prime, \(x\) is an integer and \(x^2 \equiv 1 \bmod p\) then either\[x \equiv 1 \bmod p \qquad \text{or} \qquad x \equiv -1 \bmod p\] (3)
(iii) A company has £13 940 220 to share between 11 charities.
Without performing any division and showing all your working, decide if it is possible to share this money equally between the 11 charities. (2)
Without performing any division and showing all your working, decide if it is possible to share this money equally between the 11 charities. (2)
| Scheme | Marks | AO |
|---|---|---|
| For any correct value for \(a = 4\), 6 or 12 | M1 | 1.1b |
| For all three correct values for \(a\) and no extras \(a = 4\), 6 & 12 | A1 | 1.1b |
| (2) |
Notes
M1: For an understanding of mod notation and finding a correct value for \(a = 4\), 6 or 12
A1: For all three correct values for a and no extras \(a = 4\), 6 & 12
| Scheme | Marks | AO |
|---|---|---|
| \(x^2 - 1\) is divisible by \(p\) OR \(x^2 - 1 \equiv 0 \bmod p\) OR \(p / (x^2 - 1)\) | B1 | 1.1b |
| \(\therefore (x-1)(x+1)\) is divisible by \(p\) and since \(\boldsymbol{p}\) is prime either \((x-1)\) is divisible by \(p\) or \((x+1)\) is divisible by \(p\) OR \(\therefore (x-1)(x+1) \equiv 0 \bmod p\) and since \(\boldsymbol{p}\) is prime either \(x - 1 \equiv 0 \bmod p\) or \(x + 1 \equiv 0 \bmod p\) OR \(\therefore p / (x-1)(x+1)\) and since \(\boldsymbol{p}\) is prime either \(p / (x-1)\) or \(p / (x+1)\) | M1 | 2.1 |
| \(\therefore \quad x \equiv 1 \bmod p \quad\) or \(\quad x \equiv -1 \bmod p\) * | A1* | 1.1b |
| (3) |
Notes
(ii) see scheme
| Scheme | Marks | AO | ||
|---|---|---|---|---|
For selecting and performing a divisibility test for dividing by 11
| M1 | 1.1b | ||
| Fully correct method with reason (must have correct sum \(\pm 3\)) and conclusion £13 940 220 is not divisible by 11 Therefore, it is not is it possible to share this money equally between the 11 charities | A1 | 3.2a | ||
| (2) | ||||
| (7 marks) |
Notes
M1: For applying a divisibility test for dividing by 11 to £13 940 220 or 139 402 2000p
A1: Fully correct method and concludes not divisible by 11 and interprets conclusion in context