Question

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

Answer 1

Is the statement ∃!x∈R:(x−2 =√x+7) true or false? ... Problem 8. Let x, y, and z be natural numbers. Prove or disprove: If x+y is odd and y+z is odd, then x+z is odd.

Explanation:

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Answer 2

True

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.