Question

If $a,b$ and $n$ are natural numbers then $a^{2n-1}+b^{2n-1}$ is divisible by

• A. $a+b$
• B. $a-b$
• C. $a^3+b^3$
• D. $a^2+b^2$

Answer 1

The correct option is A $a+b$

Let $P\left(n\right)=a^{2n-1}+b^{2n-1}$

$P\left(1\right)=a+b$

$P\left(2\right)=a^3+b^3=(a+b)(a^2+b^2-ab)$

Out of given options we can conclude that $P\left(n\right)$ is divisible only by $(a+b)$

It can also be done by Mathematical Induction.

But here it is objective, so verification is good enough.