Prove or disprove: For any odd integer x, (x^2 - 1) is divisible by 8.

True

Step-by-step explanation:

To prove by induction, first show that the statement is true at an initial value (in this case, x = 1).

8 | (1² - 1)

8 | 0

Next, assume the statement is true at x = k.

8 | (k² - 1)

Now show that it is true at the next value of x (x = k + 2).

(k + 2) ² - 1

= k² + 4k + 4 - 1

= k² - 1 + 4 (k + 1)

k + 1 is an even number, so 4 (k + 1) is a multiple of 8. And since we're assuming that k² - 1 is a multiple of 8, that means the sum is also a multiple of 8.

8 | ((k + 2) ² - 1)

So the statement is true.

Logically, here's another way of proving it:

x² - 1 = (x - 1) (x + 1)

Since x is an odd integer, both x - 1 and x + 1 are even numbers. Since the difference between them is 2, one of them is a multiple of 4. An even number times a multiple of 4 is a multiple of 8.