Question

Prove that x^n-Y^n divisible by x-y for all natural numbers x,y (x!=y),and n.

Answer 1

Let's do that by induction :
For
`n=1`,
`x^1-y^1` is obviously divisible by
`x-y`

If we assume the property holds at rank
`n`, then
`x^(n+1)-y^(n+1)=x(x^n-y^n)+y^n(x-y)`. Since
`x^n-y^n` is divisible by
`(x-y)`, we have
`A` such that
`x^n-y^n=A(x-y)` hence
`x^(n+1)-y^(n+1)=(x-y)(Ax+y^n)`.

Hence by induction for all
`n\ge1`,
`x-y` divides
`x^n-y^n`