Question

Let S be the set of all ordered pairs (x, y) of positive integers satisfying the condition x² - y² = 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.

Answer 1

Answer is : (b) S is the empty set

x and y are positive integer

x² - y² = 12345678

RHS 12345678 is and even number and last digit is 8.

∴ The last digit of x be 3, 7

and the last digit of y be 1, 9

 ∴x and y must be odd and square of difference is multiple of 8 but RHS is not multiple of 8.

∴ S is the empty set.