Proof：練習問題

Use the method of proof by exhaustion to determine the set of all integers \( n \in \{1, 2, 3, 4\} \) for which the expression \( n^2 - n + 11 \) yields a prime number.

Using proof by exhaustion, find the possible remainders when \( n^2 + 2 \) is divided by 4, where \( n \) is any integer.

To prove the statement "If \( n^2 \) is even, then \( n \) is even" by contradiction, which of the following is the correct assumption to make at the start of the proof?

To prove the statement "For all integers \( n \), if \( n^2 \) is a multiple of 3, then \( n \) is a multiple of 3" using the method of proof by contradiction, which of the following is the correct initial assumption to make?

Prove that for all integers \( n \), the expression \( n^2 + 3n + 2 \) is always an even number.

Prove by exhaustion that for all integers \( n \), the expression \( n^2 + 1 \) is not divisible by 3.
[5 marks]

Provide a counter-example to disprove the statement: "The expression \( n^2 + n + 1 \) is a prime number for all positive integers \( n \)."

Prove by exhaustion that for all positive integers \(n\) where \(1 \le n \le 4\), the expression \(n^2 + 3n + 2\) is always an even number.

Use the method of proof by contradiction to show that \(\sqrt{3}\) is an irrational number.