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:

Three expressions involving distinct natural numbers $m$ and $n$ need to be tested to determine which always produce integers. Distinct means $m \neq n$.

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For option (A): $m/n + n/m$

Testing with $m = 1, n = 2$:

$\frac{1}{2} + \frac{2}{1}$

$= 0.5 + 2$

$= 2.5$

This is not an integer, so (A) is not always an integer.

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For option (B): $mn(m/n+n/m)(m^2 + n^2)^{-1}$

Simplifying the fraction part:

$\frac{m}{n} + \frac{n}{m}$

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

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

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Substituting back into the original expression:

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

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

$= 1$

Option (B) always equals 1, which is an integer.

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For option (C): $mn/(m^2 + n^2)$

Testing with $m = 1, n = 2$:

$\frac{1 \times 2}{1^2 + 2^2}$

$= \frac{2}{1 + 4}$

$= \frac{2}{5}$

$= 0.4$

This is not an integer, so (C) is not always an integer.

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Only option (B) is guaranteed to always be an integer. The answer is (B) only.