Question

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

Answer 1

`x-1` is a factor of
`x^n - 1`

Step-by-step explanation:

`x-1` is a factor of
`x^n - 1`

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

For n = 1,
`x-1` is a factor
`x-1`, which is true.

Let the given statement be true for n = k that is
`x-1` is a factor of
`x^k - 1`.

Thus,
`x^k - 1` can be written equal to
`y(x-1)`, where y is an integer.

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

`x^(k+1) - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)`

Thus,
`x^k - 1` is divisible by
`x-1`.

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