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:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) only
Each expression is first written as a single fraction; it is an integer only if the denominator divides the numerator. The key fact is that for distinct natural numbers, \(m^2+n^2\gt2mn\), because \(m^2+n^2-2mn=(m-n)^2\gt0\).
(A) \(m/n + n/m\) Incorrect. \(\frac{m}{n}+\frac{n}{m}=\frac{m^2+n^2}{mn}\). After cancelling any common factor, \(mn\) can divide \(m^2+n^2\) only when \(m=n\); for example \(m=1,\ n=2\) gives \(\frac{5}{2}\), which is not an integer.
(B) \(mn (m/n+n/m) (m^2 + n^2)^{-1}\) Correct. Using the fraction from (A):
\(\displaystyle mn\times\frac{m^2+n^2}{mn}\times\frac{1}{m^2+n^2}=1\)
The value is always 1, an integer, whatever \(m\) and \(n\) are.
(C) \(mn/(m^2 + n^2)\) Incorrect. Since \(m^2+n^2\gt2mn\), the fraction \(\frac{mn}{m^2+n^2}\) is always between 0 and \(\frac{1}{2}\), so it can never be an integer.