Consider the proposition:
'If $2^n - 1$ is not prime, then $n$ is not prime' - HSC - SSCE Mathematics Extension 2 - Question 10 - 2020 - Paper 1
Question 10
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 pro... show full transcript
Worked Solution & Example Answer:Consider the proposition:
'If $2^n - 1$ is not prime, then $n$ is not prime' - HSC - SSCE Mathematics Extension 2 - Question 10 - 2020 - Paper 1
Step 1
A. $2^5 - 1$ is prime
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Answer
This statement does not disprove the proposition because if n=5, then 25−1=31, which is prime. Hence, it supports the proposition.
Step 2
B. $2^6 - 1$ is divisible by 9
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Answer
26−1=63, which is not prime. However, n=6 is not prime. This statement does not disprove the proposition because it does not provide a counterexample.
Step 3
C. $2^7 - 1$ is prime
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Answer
This statement does not disprove the proposition as 27−1=127, which is prime. Thus, it supports the original statement.
