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

1.	If xn – 1 is divisible by x – k, then the least positive integral value of k isA. 1B. 2C. 3D. 4
Answer» A. 1 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.

If xn – 1 is divisible by x – k, then the least positive integral value of k isA. 1B. 2C. 3D. 4

Answer»

A. 1

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.