Question

x^(n)+y^(n) is divisible by x+y for odd natural numer n>_1

Answer 1

consider
x^(n-1) - x^(n-2) y + ... + x y^(n-2) - y^(n-1)
Now multiply that by x+y and you have
x*n - x^(n-1) y + ... - x^2 y^(n-2) + xy^(n-1)
+x^(n-1) y - x^(n-2) y^2 + ... + x^2 y^(n-2) + y^n
= x^n + y^n

this is just an extension of sum of cubes, 5th, etc.
x^3 + y^3 = (x+y)(x^2-xy+y^2)
x^5 + y^5 = (x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)
...

Answer 2

Interesting

Answer 3

yes