Consider the proposition:
‘If $2^n - 1$ is not prime, then $n$ is not prime’ - HSC - SSCE Mathematics Extension 2 - Question 7 - 2020 - Paper 1
Question 7
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 p... 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 7 - 2020 - Paper 1
Step 1
A. $2^5 - 1$ is prime.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Calculating 25−1 gives:
25−1=32−1=31
Since 31 is prime, this statement does not disprove the original proposition.
Step 2
B. $2^6 - 1$ is divisible by 9.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Calculating 26−1 gives:
26−1=64−1=63
Now, checking divisibility:
63extmod9=0
This shows that 63 is divisible by 9, meaning n=6 (not prime) does not help disprove the proposition directly.
Step 3
C. $2^7 - 1$ is prime.
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Calculating 27−1 gives:
27−1=128−1=127
Since 127 is also prime, this statement does not disprove the original proposition.
Step 4
D. $2^{11} - 1$ is divisible by 23.
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Calculating 211−1 gives:
211−1=2048−1=2047
Now, checking divisibility:
2047extdividedby23=89
This means that 2047 is not prime since it can be factored into primes (23 and 89). Thus, this statement disproves the original proposition.
