Question

Prove that for each integer a, if a² - 1 is even, then 4 divides a² - 1 by definition.

a. Proof by Contradiction
b. Proof by Induction
c. Direct Proof
d. Proof by Counterexample

Answer 1

To prove that for each integer a, if a² - 1 is even, then 4 divides a² - 1, we can use a direct proof method.

Step-by-step explanation:

To prove that for each integer a, if a² - 1 is even, then 4 divides a² - 1, we can use a direct proof method.

Let's assume that a is an integer such that a² - 1 is even. This means that a² - 1 is divisible by 2.

We can express a² - 1 as (a - 1)(a + 1). Since a² - 1 is divisible by 2, one of the factors (a - 1) or (a + 1) must be divisible by 2.

Let's consider two cases:

• If (a - 1) is divisible by 2, then a is an odd integer. In this case, (a + 1) must be an even integer, and 2 divides (a + 1).
• If (a + 1) is divisible by 2, then a is an even integer. In this case, (a - 1) must be an odd integer, and 2 divides (a - 1).

Thus, in both cases, 2 divides either (a - 1) or (a + 1). Therefore, 4 divides a² - 1 by definition.