Final answer:
To prove the statement 'For every integer n, if n is odd, then 8 | (n² - 1)', you can use a direct proof, indirect proof, contrapositive proof, or proof by contradiction.
Step-by-step explanation:
To prove the statement 'For every integer n, if n is odd, then 8 | (n² - 1)' using each of the given methods:
a) Direct proof: Assume n is an odd integer. Let n = 2k+1, where k is an integer. Substituting this into the expression n²-1, we get (2k+1)²-1 = 4k²+4k = 4k(k+1). Since k and (k+1) are consecutive integers, one of them must be even. Let's say k is even, so k+1 is odd. Therefore, 4k(k+1) is divisible by 8.
b) Indirect proof: Assume n is an odd integer and 8 is not divisible by (n² - 1). This means (n² - 1) is not divisible by 8. Follow the steps in the direct proof to show a contradiction.
c) Contrapositive proof: The contrapositive of the statement is 'For every integer n, if 8 is not divisible by (n² - 1), then n is even'. Assume 8 is not divisible by (n² - 1). Follow the steps in the direct proof to show that n must be even.
d) Proof by contradiction: Assume n is an odd integer and also assume that 8 is not divisible by (n² - 1). Show that these assumptions lead to a contradiction by following the steps in the direct proof.