證明：練習題

Which of the following is a counter-example to the statement:
"If \( n \) is a positive integer, then \( 2n^2 + 1 \) is always a prime number."

Which of the following expressions is always even for any integer \( n \)?

In the proof by contradiction that \( \sqrt{2} \) is irrational, we assume \( \sqrt{2} = \frac{p}{q} \) where \( p \) and \( q \) are integers with no common factors. After demonstrating that both \( p \) and \( q \) must be even, which part of the initial assumption is directly contradicted?

Consider the statement: "The sum of two irrational numbers is always irrational."
Which pair of numbers provides a counter-example to this statement?

Which of the following is the correct initial assumption to prove the following statement by contradiction?
"If \( n^3 \) is an even integer, then \( n \) is an even integer."

A student attempts to prove that the sum of any three consecutive integers is always a multiple of 3. They begin by letting the three integers be \( n-1 \), \( n \), and \( n+1 \).
Show the algebraic steps required to complete this proof by deduction.

Prove by contradiction that if \( n^2 \) is a multiple of 3, then \( n \) must be a multiple of 3.
(You may assume that any integer not divisible by 3 can be written in the form \( 3k+1 \) or \( 3k+2 \)).

When proving by contradiction that "the sum of a rational number and an irrational number is always irrational", what is the necessary initial assumption?

To prove that "There is no greatest even integer" using proof by contradiction, a student starts with a specific assumption.

(a) State the initial assumption the student must make to begin this proof.

(b) Following this assumption, let the greatest even integer be \( M \). By considering the number \( M + 2 \), complete the argument to reach a contradiction.

Consider the statement: "If \( n^2 \) is a multiple of 3, then \( n \) is a multiple of 3."

To prove this by contradiction, a student begins by assuming the negation of the statement.

(a) State clearly the initial assumption the student should make.

(b) Complete the proof to show that if \( n \) is not a multiple of 3, then \( n^2 \) cannot be a multiple of 3. You may assume that any integer not divisible by 3 can be written in the form \( 3k+1 \) or \( 3k+2 \) for some integer \( k \).