Moderate -0.8 This is a straightforward proof by contradiction requiring only basic logic about integers. Students need to assume a greatest multiple exists, then show 5 times that number is larger, which is a standard textbook exercise with minimal steps and no sophisticated mathematical insight required.
Assume that there is a greatest multiple of 5 ie \(N=5k\)
B1*
Assumption for contradiction; some indication that they are starting with the greatest multiple of 5
\(N+5=5k+5=5(k+1)\)
M1
Add on 5, or a multiple of 5; or any equiv operation that would result in a larger multiple of 5; M0 if just numerical example
This is a multiple of 5, and \(N+5>N\) which contradicts the assumption. Hence there is no greatest multiple of 5
A1d*
Statement denying assumption; need justification about why it is a multiple of 5, why it is greater, as well as 'contradiction' or clear equiv such as 'initial assumption is incorrect'
[3]
# Question 4:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Assume that there is a greatest multiple of 5 ie $N=5k$ | B1* | Assumption for contradiction; some indication that they are starting with the greatest multiple of 5 |
| $N+5=5k+5=5(k+1)$ | M1 | Add on 5, or a multiple of 5; or any equiv operation that would result in a larger multiple of 5; M0 if just numerical example |
| This is a multiple of 5, and $N+5>N$ which contradicts the assumption. Hence there is no greatest multiple of 5 | A1d* | Statement denying assumption; need justification about why it is a multiple of 5, why it is greater, as well as 'contradiction' or clear equiv such as 'initial assumption is incorrect' |
| **[3]** | | |
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