June 2022 Paper 2 Q5
5 Tom conjectures that if \(n\) is an odd number greater than 1, then \(2^n - 1\) is prime.
Find a counter example to disprove Tom’s conjecture. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(2^n - 1\) correctly evaluated for any odd positive integer | B1 | 1.1 |
| \(2^n - 1\) correctly evaluated for any odd positive integer for which Tom’s conjecture is false | B1 | 2.1 |
| eg 511 is divisible by 7 with 9 seen [so not prime] | B1 | 2.2a |
| [3] |
Notes
B1: \(n \geqslant 3\)
B0 if only rounded number in standard form seen
B1: eg \(2^9 - 1 = 511\), eg \(2^{15} - 1 = 32767\) eg \(2^{21} - 1 = 2097151\)
B1: NB 32767 and 2097151 both divisible by 7;
2047 divisible by 23
correct value of \(n\) may be embedded in formula
NB B0 if answer spoiled by eg so 511 is prime