Question

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

• A. $S$ is an infinite set
• B. $S$ is the empty set
• C. $S$ has exactly one element
• D. $S$ is finite set and has atleast two elements

Answer 1

The correct option is B $S\text{is the empty set}$

Given:
$x^2–y^2=12345678\Rightarrow (x+y)(x-y)=12345678$
As $x$ and $y$ are positive integers.
$x+y$ will be positive, so $x-y$ should be positive.
As the RHS is even number, so one of them should be even.
If $(x+y)$ is even then $(x-y)$ wil also be even and vice-versa.
Therefore both $(x+y)$ and $(x-y)$ are even.
So $(x+y)(x-y)$ will be divisible by 4, but RHS is not divisible by 4.
Hence there does not exist any pair of $(x,y)$.

Hence, The correct answer is option B.