Q25 / 54 · KVPY 2017 (5 Nov SA) · Mathematics

Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers satisfying the condition $x^{2}-y^{2}=12345678$. Then

  • A
    $S$ is an infinite set
  • Correct answer
    B
    $S$ is the empty set
  • C
    S has exactly one element
  • D
    $S$ is a finite set and has at least two elements.

Solution

$x^{2}-y^{2}=12345678\left(x, y \in I^{+}\right)$ R.H.S. is even, so $\mathrm{x}$, y should be odd integer but difference of square of two odd integers is multiple of 8 but R.H.S. is not multiple of 8

Hence, the correct answer is option: 'B'.

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