Consider the proposition:
‘If $2^n - 1$ is not prime, then $n$ is not prime$^2$ - 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$^2$.
Given that each of the following statements is true, which statement disproves the... show full transcript
Worked Solution & Example Answer:Consider the proposition:
‘If $2^n - 1$ is not prime, then $n$ is not prime$^2$ - 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 indicates that when n=5, 25−1=31, which is indeed a prime number. However, this does not disprove the original proposition, as it confirms the case where n can be prime and still have 2n−1 as prime.
Step 2
B. $2^6 - 1$ is divisible by 9
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Answer
Calculating 26−1 gives us 63, which is divisible by 9. This means that when n=6, 26−1 is not a prime number (since it's composite). However, n=6 is not a prime number either, which does not contradict the original statement.
Step 3
C. $2^7 - 1$ is prime
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Answer
Evaluating this, we find that 27−1=127, which is a prime number. Like in option A, this does not invalidate the proposition, since when n=7, it upholds the condition of the statement.
