Moderate -0.8 Part (i) requires finding a simple counter-example (n=3 gives 3³+2=29 which is prime, but n=5 gives 3⁵+2=245=5×49 which is not prime). Part (ii) is a standard proof by exhaustion checking m≡1,2 (mod 3), both routine calculations. Both parts are straightforward applications of basic proof techniques with minimal problem-solving required.
7. (i) Prove by counter example that the statement
"If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number."
is false.
(ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then
$$m ^ { 2 } - 1$$
is divisible by 3
(3)
(Total for Question 7 is 5 marks)
7. (i) Prove by counter example that the statement\\
"If $n$ is a prime number then $3 ^ { n } + 2$ is also a prime number."\\
is false.\\
(ii) Use proof by exhaustion to prove that if $m$ is an integer that is not divisible by 3 , then
$$m ^ { 2 } - 1$$
is divisible by 3\\
(3)\\
(Total for Question 7 is 5 marks)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q7 [5]}}