Challenging +1.2 This requires understanding proof by contradiction and identifying logical gaps in a classic proof, but the correct proof structure is well-known and the error is straightforward (the new number need not be prime itself, only have a prime factor not in the list). It's above average difficulty due to requiring proof-writing skills and conceptual understanding rather than calculation, but not exceptionally challenging for Further Maths students.
5 A student's attempt to prove by contradiction that there is no largest prime number is shown below.
If there is a largest prime, list all the primes.
Multiply all the primes and add 1.
The new number is not divisible by any of the primes in the list and so it must be a new prime.
The proof is incorrect and incomplete.
Write a correct version of the proof.
Multiply all the primes up to and including \([p]\) and add 1. This number is not divisible by any of the primes
Therefore it is prime (and larger than \(p\))
A1
This is a contradiction (so there is no largest prime)
A1
Must mention 'contradiction'
[3]
## Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Suppose there is a largest prime, $[p]$ | M1 | Allow 'If there is …' or equivalent |
| Multiply all the primes up to and including $[p]$ and add 1. This number is not divisible by any of the primes | | |
| Therefore it is prime (and larger than $p$) | A1 | |
| This is a contradiction (so there is no largest prime) | A1 | Must mention 'contradiction' |
| **[3]** | | |
---
5 A student's attempt to prove by contradiction that there is no largest prime number is shown below.\\
If there is a largest prime, list all the primes.\\
Multiply all the primes and add 1.\\
The new number is not divisible by any of the primes in the list and so it must be a new prime.
The proof is incorrect and incomplete.\\
Write a correct version of the proof.
\hfill \mbox{\textit{OCR MEI Paper 3 2019 Q5 [3]}}