Question

How many pairs of integers (x,y) are there such as x²y-xy²=3015

Answer 1

That's how I started the question: \(xy(x-y)=3015\\xy\neq0,\\x-y\neq0,\\x\neq0,\\y\neq0\)

Given that are there algebraic solutions?

Answer 2

Further factorisation gives \(xy(x-y)=3^{2}*5*67\)
I guess it's by trial and error, then?

Answer 3

i was doing the same thing

Answer 4

but i guess that would be lengthy :/

Answer 5

normally i isolate one of the given unknowns and write it in terms of the other unknown and then i use the information that "both x and y are integers" to find the possible combinations

Answer 6

This would be impractical, I guess. I'd give a try doing something about the prime factorization and the given conditions. But dunno where to start :(

Answer 7

Well, I have the alternatives though:
(A) 1005
(B) 603
(C) 8
(D) 1
(E) 0

Answer 8

yeah isolating them seems impractical
i'm trying to do some useful substitutions for x and y
like rn i'm trying to convert it into a useful form by substituting x=t+1 and y=p+1

Answer 9

http://www.wolframalpha.com/input/?i=solve+x%C2%B2y-xy%C2%B2%3D3015+over+integer+values. It uses quadratic formula/completing the square. But the answer is not convicing

Answer 10

yeah i totally forgot that thing uh
we can assume it to be a quadratic in y
and then by using quadratic formula you can get the values of y
remember that y is an integer so the discriminant must be a perfect square
so using this you get the solution pairs

Answer 11

Hint : x and y must be odd

Answer 12

Yeah, given the factors.

Answer 13

xy(x-y) = 3015

Since 3015 is odd, we can say x, y, and x-y must be odd

Answer 14

When x and y are odd, what can we say about x-y ?

Answer 15

There's no solutions for x-y being odd. Now that's a simple way!

Answer 16

This proof is good enough. xy will be odd, but x-y will be always even. GIven that, xy*(x-y) will always be even.

Answer 17

Looks nice and simple to me!

Answer 18

Thanks, saved me a bunch of impractical stuff :D

Answer 19

@ChillOut from this you should learn to always trust wolframalpha ;)

Good question. Nice job @ganeshie8

Answer 20

@agent0smith I trust it and I know it's right. Hell, I use it a lot! I meant It wasn't a proof that should be used :D

Answer 21

haha i know what you meant :D