Q28 of 30 Page 70

If xn – 1 is divisible by x – k, then the least positive integral value of k is

Given; P(n): xn – 1 is divisible by x – k

⇒ P(1) : x – 1


⇒ P(2) : x2 – 1 = (x−1)(x+1)


⇒ P(3) : x3 – 1 = (x−1)(x2+xy+1)


⇒ P(4) : x4 − 1 = (x2−1)(x2+1) = (x−1)(x+1)(x2+1)


∴ The least positive integral value of k is 1.

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