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1.Set Theory
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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
B
$S$ is the empty set
C
$S$ has exactly one element
D
$S$ is a finite set and has at least two elements.
(KVPY-2017)
Solution
(b)
$x$ and $y$ are positive integer
$x^2-y^2=12345678$
$RHS$ $12345678$ is and even number and last digit is $8 .$
$\therefore$ The last digit of $x$ be $3,7$
and the last digit of $y$ be $1,9$.
$\therefore x$ and $y$ must be odd and square of difference is multiple of $8$ but $RHS$ is not multiple of $8$ .
$\therefore S$ is the empty set.
Standard 11
Mathematics
