JIPMAT 2023 (QA) - <p>If k<sup>2</sup> - 1 is divisible by 8, then K is:</p> | PYQs + Solutions

JIPMAT 2023

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:

k^2 - 1

is divisible by 8, then:

\newline

k^2 - 1 = 8m

for some integer

m

k^2 = 8m + 1

Since

k^2 = (k+1)(k-1) + 1

, we can write:

\newline

(k+1)(k-1)

must be divisible by 8

For consecutive integers

(k+1)

and

(k-1)

\newline

- If

k

is even, one number is odd and other is odd

\newline

- 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.