Standard +0.3 This question requires students to test small odd values systematically (n=3,5,7,9...) and recognize that 2^9-1=511=7Γ73 is composite. While it involves some calculation and understanding of counterexamples, it's a straightforward 3-mark question requiring only basic arithmetic and no sophisticated number theory.
Question 5:
5 | 2π β1 correctly evaluated for any odd
positive integer
2π β1 correctly evaluated for any odd
positive integer for which Tomβs conjecture
is false
eg 511 is divisible by 7 with 9 seen [so not
prime] | B1
B1
B1 | 1.1
2.1
2.2a | n β₯ 3
B0 if only rounded number in standard form seen
eg 29 β1 = 511, eg 215 β1 = 32767 eg 221 β1 = 2097151
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
[3]
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]
\hfill \mbox{\textit{OCR MEI Paper 2 2022 Q5 [3]}}