Prove that x-1 is a factor of x^n-1 for any positive integer n.

is a factor of

Step-by-step explanation:

is a factor of

We will prove this with the help of principal of mathematical induction.

For n = 1, is a factor , which is true.

Let the given statement be true for n = k that is is a factor of .

Thus, can be written equal to  , where y is an integer.

Now, we will prove that the given statement is true for n = k+1

Thus, is divisible by .

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.