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Question

Let S be the set of all ordered pairs (x,y) of positive integers satisfying the condition
x2ā€“y2=12345678. Then

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

The correct option is B S is the empty set
Given:
x2y2=12345678(x+y)(xy)=12345678
As x and y are positive integers.
x+y will be positive, so xy should be positive.
As the RHS is even number, so one of them should be even.
If (x+y) is even then (xy) wil also be even and vice-versa.
Therefore both (x+y) and (xy) are even.
So (x+y)(xy) 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.

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