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.

yes hope im right

Step-by-step explanation: