# How to solve this Diophantine Equation. Step by Step

What are all the solutions to the diophantine equation: 6x-6y-xy=0.

-
Like question, like answer... Here it is: wolframalpha.com/input/?i=solve+diophantine+6x-6y-xy%3D0 (Press More solutions until you get all of them.) – Vadim Apr 24 '12 at 3:19
Mily, you might find more favor with the local crowd if you show some of your own effort. As Bill has noticed, a good starting point is to add $36$ to both sides and factor by grouping. – The Chaz 2.0 Apr 24 '12 at 3:48
Thanks for your help guys. I'm creating a program designed to solve Diophantine Eq. I just want to know how real solvers would react to a plain question, and yes, the solution is group the equation so that we have a product of two terms equal to 36. Then, by using prime decomposition , it'll be easy to find all possible solutions for integers values of x and y. Thank you all. Mily – Mily Apr 24 '12 at 4:10
You might be interested in emailing the author of this Diophantine solver for suggestions! – The Chaz 2.0 Apr 24 '12 at 4:21
I presume that you are the user who posted the question as well, so I flagged the moderators to merge the two accounts. – Srivatsan Apr 24 '12 at 4:40

Hint $\$ Rewrite it as $\rm\: (6+x)(6-y) = 36.$

-

The 18 integer solutions are :

x = -42, y = 7

x = -24, y = 8

x = -18, y = 9

x = -15, y = 10

x = -12, y = 12

x = -10, y = 15

x = -9, y = 18

x = -8, y = 24

x = -7, y = 42

x = -5, y = -30

x = -4, y = -12

x = -3, y = -6

x = -2, y = -3

x = 0, y = 0

x = 3, y = 2

x = 6, y = 3

x = 12, y = 4

x = 30, y = 5

-