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If m and n are direct natural numbers, then which of the following is/are integer/integers?
(1) m/n + n/m
(2) mn × (m/n + n/m) × (m2 + n2)-1
(3) mn/ (m2 + n2)
Select the correct answer using the code given below. 
  • a)
    1 and 2
  • b)
    2 only
  • c)
    2 and 3
  • d)
    3 only
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If m and n are direct natural numbers, then which of the following is/...
Statement I:
m/n + n/m = (m2 + n2) /mn
Let this be an integer k
⇒ m2 + n2 = kmn
⇒ (m2 + n2 – 2mn) = (k – 2) × mn
⇒ (m – n)2 = (k – 2) × mn
⇒ m – n = √[(k – 2) × mn]
⇒ √[(k – 2) × mn] for a integer, requires a specific combination of (m, n)
For example, if m = 1 and n = 2
⇒ ½ + 2 = 5/2 not an integer
Statement II:
mn × (m/n + n/m) × (m2 + n2)-1
⇒ mn × [(m2 + n2) /mn] × (m2 + n2)-1 = 1
⇒ It is an integer
Statement III:
It is the reciprocal of the expression in statement I
When m = 1 and n = 2
The value will be 2/5 which is not an integer
∴ Only statement 2 is correct.
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If m and n are direct natural numbers, then which of the following is/are integer/integers?(1) m/n + n/m(2) mn × (m/n + n/m) × (m2+ n2)-1(3) mn/ (m2+ n2)Select the correct answer using the code given below.a)1 and 2b)2 onlyc)2 and 3d)3 onlyCorrect answer is option 'B'. Can you explain this answer?
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