If k 2 - 1 is divisible by 8, then K is: Option 1: an integer Option 2: a natural number Option 3: an even integer Option 4: an odd integer

Answer: an odd integer. Solution: Let me solve this: If k^2 - 1 is divisible by 8, then: k^2 - 1 = 8m for some integer m k^2 = 8m + 1 Since k^2 = (k+1)(k-1) + 1, we can write: (k+1)(k-1) must be divisible by 8 For consecutive integers (k+1) and (k-1): - If k is even, one number is odd and other is odd - If k is odd, one number is even and other is even For their product to be divisible by 8, k must be odd (so both (k+1) and (k-1) are even) Therefore, k must be an odd integer. Alternatively, you can just put your own values for k according to the options and find out.

Number System

Divisibility Rules

Conceptual

If k² - 1 is divisible by 8, then K is:

an integer

a natural number

an even integer

an odd integer

✅ Correct Option: 4

Let me solve this:

If $k^2 - 1$ is divisible by 8, then:

$k^2 - 1 = 8m$ for some integer $m$

$k^2 = 8m + 1$

Since $k^2 = (k+1)(k-1) + 1$ , we can write:

$(k+1)(k-1)$ must be divisible by 8

For consecutive integers $(k+1)$ and $(k-1)$ :

If $k$ $k$ is even, one number is odd and other is odd
- If $k$ is odd, one number is even and other is even

For their product to be divisible by 8, $k$ must be odd (so both $(k+1)$ and $(k-1)$ are even)

Therefore, $k$ must be an odd integer.

Alternatively, you can just put your own values for k according to the options and find out.

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If k2 - 1 is divisible by 8, then K is:

Solution

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If k² - 1 is divisible by 8, then K is: