elem.num.theo - diophantine equations
define all solutions in integers to diophantine equations:
56x+72y=40
ok - I'll write my steps, my problem is that I end up with no solutions at all - maybe someone can find where I went wrong ... ?
SO:
72 = 56(1) + 16
56 = 16(3) + 8
16 = 8(2)
so gcd(56,72) = 8
next:
8 = 56 - (16)(3)
= 56 - 3(72-56)
= 56(4) + 72(-3)
40 = 56(20) + 72(-15)
so x0=20 & y0=-15
so:
x = 20 + 9t, for some int. t
y = -15 - 7t, for some int. t
and I get:
-(15/7) > t > -(20/9)
but there are no integers that work.
did I do something stupid in algebra or can there really be no solutions?