which of the following statements is true?

Question

which of the following statements is true of the number of pairs of positive integers (x,y) satisfying the eqn x^2-y^2=702 ?

(1)There is only one pair possible

(2)The number of pairs is more than one but finite

(3)There are infinite number of pairs

(4)No such pair is possible

Answer 1

X^2 = 702 + y^2
(4).

While solving the for x based upon integer Y's.
When Y > 350 the solutions of X are less than Y + 1 implies that there cannot be an integer solution with Y > 350 and there were no solutions if integer pair of X,Y less than 350.

Answer 2

There are an infinite number of solution pairs. But are they integers?

Try just a few:
Y=0, so X = sqrt(702)
Y=1, X = sqrt(703)
X=0, Y = sqrt(-702)...not a real, positive value.
Y = infinity, X = sqrt(702 + infinity^2) = infinity.

Between Y=0 and Y=1 alone, there are an infinite number of possible solutions, but they won't be integers.

And, we haven't even talked about the negative or complex (imaginary number) solutions yet!

The equation describes an hyperbola, which goes on to infinity.

The real challenge is finding the integer pairs....

It looks like the answer is (4) No such pair is possible.

Answer 3

There are no integers x and y such that x^2 - y^2 = 702. Statement (4) is the true one.

Answer 4

x>y, because x^2 - y^2 > 0

So x - y > 0

(x+y)(x-y)=702

702=2 * 3^3 * 13

702 has 16 factors.

The numbers of pairs ( x+y , x-y ) is 16 C 2 = 120

The number of pairs (x,y) is the same: 120

So the answer is (2)

Answer 5

(3) for all the reasons in Steve W's answer.

Considering only the real solutions, the equation is a parabola. Since parabolas contain infinitely many points, there are infinitely many (x,y) pairs that satisfy the equation.

Answer 6

My dog's breath smells like cat food.