How many pairs of positive integers $(x,y)$ are there which lie below the graph of the hyperbola $xy = 16$?

Answer 1

infinitely many.

Or did you want to restrict the available area?

In the first quadrant, (4,4) is on the curve.

There will be none to the left of (1,16) or to the right of (16,1).

And don't count the lattice points on the curve...

Answer 2

restrict

Answer 3

do you see what a useless response that is? Restrict how?

Anyway, I gave you the information you need.

How about showing some effort of your own here?

Answer 4

its 45 when you count it

Answer 5

Its not infinite. Its finite. Basically, its 0<xy<16.

Answer 6

The answer 45 is correct. You simply find all pairs of integers x and y that multiply to a product greater than zero and less than sixteen.

Answer 7

The problem is restricted to x and y being positive integers - no other restrictions are needed. You can allow an infinite area, the answer is still 45.

Answer 8

Hmm... I got 46. Though I do think 45 is correct; I must've had a miscalculation. Nice explanation, Overseer! :)

Answer 9

Oh, I just found a new way to solve it. And wow, this is an old problem :')

But you can just put x in the equation for everything between 1 and 16, then find the y values and add them up. Got me 45!

Answer 10

Yes, the answer is 45