Foundation November 2024 Paper 1 Q18
18
(a) \(N\) is a number such that:
- \(N = 3 \times 5 \times k\), where \(k\) is a prime number
- \(N\) is greater than 400.
Find the smallest possible value of \(N\). [3]
(b) \(a\) and \(b\) are different prime numbers.
Explain why \(a \times b\) is not a prime number. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 435 | 3 | B2 for [\(k =\)] 29 or for an answer that satisfies both conditions e.g. \(3 \times 5 \times 31 = 465\) or M1 for \(400 \div (3 \times 5)\) maybe implied by 26.66… or 26.7 or for \(3 \times 5 \times\) (their prime number) correctly evaluated or for \(3 \times 5 \times 27 = 405\) | e.g. 29 used in final trial or even as their answer using any prime number that is greater than 29 e.g. M1 for \(3 \times 5 \times 23 = 345\) in working or \(3 \times 5 \times 31 = 465\) with 31 as the answer treat factor trees or factor tables as multiplication |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| any correct reason e.g. \(a\) and/or \(b\) are factors | 1 | see appendix | |
Appendix
Exemplar responses for 18b
| Response | Mark |
|---|---|
| it has four factors [1, \(a\), \(b\), \(ab\)] | 1 |
| \(a\) and/or \(b\) are factors | 1 |
| [two different prime numbers multiplied together do not make a prime number] because they are both factors of the new number | 1 |
| Any non-identical prime numbers create a non-prime number when multiplied together because the resulting number can be divided by both prime numbers | 1 (BOD) |
| The product of two prime numbers is always divisible by both prime numbers …. | 1 (BOD) |
| \(a \times b\) has other factors aside from 1 and itself (\(a\) and \(b\)) | 1(BOD) |
| \(a \times b = 15\) 15 is divisible by 5 (may use other numbers) | 1 (BOD) |
| it has factors besides itself and 1 | 0 (not enough) |
| two prime numbers multiplied together do not always become a prime number | 0 |
| Two prime numbers can multiply to make a number that isn’t prime | 0 |
| 2 primes make a non prime | 0 |
| A prime number x a prime number doesn’t equal prime | 0 |
| A prime number can’t have factors that are also prime | 0 |
| It can then be a multiple of another number | 0 |
| Divisible by more numbers other than 1 and itself | 0 |