Question

Show that p – 1 is a factor of p¹⁰ – 1 and also of p¹¹ – 1.

Answer 1

Let g(p) = p¹⁰ – 1  ...(i)

And h(p) = p¹¹ – 1   ...(ii)

On putting p = 1 in equation (i), we get

g(1) = 1¹⁰ – 1

= 1 – 1

= 0

Hence, p – 1 is a factor of g(p).

Again, putting p = 1 in equation (ii), we get

h(1) = (1)¹¹ – 1

= 1 – 1

= 0

Hence, p – 1 is a factor of h(p).