Show that p-1 is a factor of p10 -1 and also of p11 -1.
Question:

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

Solution:

Let g (p) = p10 -1

and h(p) = p11 -1.

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

g(1)=110-1= 1-1=0 Hence, p-1 is a factor of g(p).

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

h (1) = (1)11 —1 = 1 —1 = 0 Hence, p -1 is a factor of h(p).