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  • n square – 1 divisible by 8, if n is (A) an integer (B) a natural number (C) an odd integer (D) an even integer

n2 – 1 divisible by 8, if n is

(A) an integer
(B) a natural number
(C) an odd integer
(D) an even integer

Answers (1)

let n = 2 then n2 – 1 = 4 – 1 = 3, which is not divisible by 8
When n is a natural number,
let n = 4 then n2 – 1 = 16 – 1 = 15, which is not divisible by 8
When n is an odd integer then n2 – 1 is always divisible by 8,
let n = 3, n2 – 1 = 9 – 1 = 8 which is divisible by 8.
Proof:  let n = 2q + 1
n2– 1 = (2q + 1)2 – 1

           = 4q2 + 1 + 4q – 1

            = 4q (q + 1)

Product of two consecutive number is divisible by 2, hence number n2 – 1 is divisible by 8.
When n is an even integer then n2 – 1 is not divisible by 8.
let n = 2,
n2 – 1 = 4 – 1 = 3

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