Consider the proposition: ‘If $2^n - 1$ is not prime, then $n$ is not prime.’ Given that each of the following statements is true, which statement disproves the proposition? A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2020 - Paper 1

Calculating this, we find:

$2^6 - 1 = 64 - 1 = 63$

Now checking for divisibility:

$63 ext{ mod } 9 = 0$

Since this is divisible by 9, this statement does hold but does not disprove the proposition.

Calculating this, we find:

$2^7 - 1 = 128 - 1 = 127$

Since 127 is prime, this statement also holds but does not disprove the proposition.

Calculating this, we get:

$2^{11} - 1 = 2048 - 1 = 2047$

To check for divisibility by 23:

$2047 ext{ mod } 23 = 0$

Since $2^{11} - 1$ is divisible by 23, and the proposition states that if $2^n - 1$ is not prime, then $n$ must not be prime. Here, $n = 11$ is prime but $2^{11} - 1$ is not prime as it is divisible by 23. Therefore, this statement disproves the original proposition.