Find two integer solutions to each of the following, or state why no solutions exist:
Starting from the bottom and substituting for the previous remainder,
We find that 31 ⋅ 7 + 27 ⋅ (-8) = 1, so 31x - 27y = 11 has an initial solution of \mathbf{{x}_{0} = 7 ⋅ 11 = 77} and \mathbf{{y}_{0} = -1 ⋅-8 ⋅ 11 = 88}.
The general solutions have the form
Another solution is given by \mathbf{x(1) = 77 - 27 ⋅ 1 = 50} and \mathbf{y(1) = 88 - 31 ⋅ 1 = 57}.