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 $2^6 - 1$, we have:

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

Now, checking divisibility by 9:

$63 ext{ is divisible by } 9$

This means $2^6 - 1$ is not prime, but it does not provide a definitive disproof of the proposition.

Calculating $2^7 - 1$, we have:

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

Since 127 is also a prime number, this statement does not disprove the proposition.

Calculating $2^{11} - 1$, we find:

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

Now, checking the divisibility by 23:

$2047 ext{ is divisible by } 23 ext{, since } 2047 = 23 imes 89$

This indicates that $2^{11} - 1$ is not prime, yet $n = 11$ is prime. Therefore, this statement disproves the proposition.