Question

Which of the following is divisible by $x^{2}-y^{2}, \forall x\ne y$?

• A. $x^{n}-y^{n} \forall\,\,\, n\in N$
• B. $x^{n}+y^{n} \forall\,\,\, n\in N$
• C. $(x^{n}-y^{n}) \, (x^{2n+1}+y^{2n+1}) \forall\,\,\, n\in N$
• D. $(x^{n}-y^{n}) \, (x^m+y^m) \forall m, n\in N$

Answer 1

Answer: (C)

$ \because 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 $\left(x^{2 n+1}+y^{2 n+1}\right)$ is divisible by $(x+y)$.