Question

X power n - y power n is divisible by x-y where X and Y belongs to n and X is not equal to y

Answer 1

Let P(n) : xⁿ – yⁿ is divisible by x – y,
where x and y are any integers with x≠y.

Now, P(l): x¹ -y¹ = x-y, which is divisible by (x-y)
Hence, P(l) is true.

Let us assume that, P(n) is true for some natural number n = k.
P(k): x^k -y^k is divisible by (x – y)
or x^k-y^k = m(x-y),m ∈ N …(i)

Now, we have to prove that P(k + 1) is true.
P(k+l):x^k+l-y^k+l
= x^k-x-x^k-y + x^k-y-y^ky
= x^k(x-y) +y(x^k-y^k)
= x^k(x – y) + ym(x – y) (using (i))
= (x -y) [x^k+ym], which is divisible by (x-y)
Hence, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for any natural number n.