Q43 / 160 · AP EAMCET 2019 (21 Apr Shift 2) · Mathematics

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

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

Solution

The expression $x^{n} - y^{n}$ 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 $(x + y) .$

Hence, the correct answer is option: 'C'.

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