If m and n are distinct natural numbers, then which of the following is/are integer(s)?

(A) $m/n + n/m$
(B) $mn (m/n+n/m) (m^2 + n^2)-1$
(C) $mn/(m^2 + n^2)$

Choose the correct answer from the options given below:

(B) only

The correct answer is Option (2) → (B) only

Given that m and n are distinct natural numbers.

(A) $\frac{m}{n} + \frac{n}{m}$​

$\frac{m}{n} + \frac{n}{m} = \frac{m^2 + n^2}{mn}$

Since $m≠n$ and both are natural numbers,
$m^2 + n^2$ is not generally divisible by mn.

Not always an integer

(B) $mn\left(\frac{m}{n} + \frac{n}{m}\right)(m^2+n^2)^{-1}$

First simplify:

$mn\frac{m}{n} + \frac{n}{m} = \frac{m^2+n^2}{mn}$

Substitute:

$mn \times \frac{m^2+n^2}{mn} \times \frac{1}{m^2+n^2}$

Everything cancels out:

=1

Always an integer

(C) $\frac{mn}{m^2+n^2}$​

Since $m^2+n^2 > mn$ for distinct natural numbers,

$\frac{mn}{m^2+n^2}$​

is a proper fraction, not an integer.

Not an integer

Correct Answer: (B) only