Which of the following is divisible by x2-y2, ∀ x≠ y?

Question

Which of the following is divisible by x2-y2, ∀ x≠ y? (A) xn-yn ∀ n∈ N (B) xn+yn ∀ n∈ N (C) (xn-yn) (x2n+1+y2n+1) ∀ n∈ N (D) (xn-yn) (xm+ym) ∀ m, n∈ N

Tardigrade Admin · Accepted Answer

∵ xn-yn is divisible by (x+y)(x-y), if n= even Similarly, xn-yn is divisible by (x-y), if n= odd and (x2 n+1+y2 n+1) is divisible by (x+y).

Q. Which of the following is divisible by $x^{2} - y^{2}, \forall x \neq = y$ ?

$x^{n} - y^{n} \forall n \in N$

$x^{n} + y^{n} \forall n \in N$

$(x^{n} - y^{n}) (x^{2 n + 1} + y^{2 n + 1}) \forall n \in N$

$(x^{n} - y^{n}) (x^{m} + y^{m}) \forall m, n \in N$

$∵ x^{n} - y^{n}$ is divisible by $(x + y) (x - y)$ , if
$n =$ even
Similarly, $x^{n} - y^{n}$ is divisible by $(x - y)$ ,
if $n =$ odd and $(x^{2 n + 1} + y^{2 n + 1})$ is divisible by $(x + y)$ .

Q. Which of the following is divisible by x2−y2,∀x=y?

xn−yn∀n∈N

xn+yn∀n∈N

(xn−yn)(x2n+1+y2n+1)∀n∈N

(xn−yn)(xm+ym)∀m,n∈N