Diophantine Africans

Since /b/ is of no help:

How do I solved ６３ｘ　＋　１０８ｙ　＝　７６５, with x and y being natural numbers? I know that a solution can be with x = 7 and y = 3, but how do you find this?

I've found the greatest common divisor, which is 9, and solving the equation gave me 108*2 + 63*(-4) = 9

However, what the fuck? Multiplying it by (765/9) doesn't make sense. 108*44-63*88 = 765?

What I have so far:

108-1*63=45
63-1*45=18
45-2*18=9

9|18

45-2*18=9
45-2*(63-1*45)=9
(108-1*63)-2*63+(108-1*63)=9
108*2+63*(-4)=9

What is I doins rong?

divide whole thing by 9 and then:
7x+12y=85
then fucking solve it by giving random values to y (you have fewer variants)
y=0 -> x=12.7 nope.
y=1 -> x=10.4 nope.
y=2 -> x=8.71 nope.
y=3 -> x=7    yesh.
etc.
and y=3 x=7 is the only right answer.

Are you a troll?

http://en.wikipedia.org/wiki/Extended_euclidean_algorithm

108*2 + 63*(-4) = -36 =/= 9

How are Africans involved?