Question

HE

Answer 1

I'm not sure why they threw in the +AB part when clearly C = A+B gives you solutions of x = 1 and y = 1

Answer 2

Well (1,1) is a solution to Ax+By = C when C = A+B

If C = A+B+AB, then this number is larger than A+B

So if there is a solution to Ax+By = A+B+AB, then it must be a positive integer ordered pair since having one coordinate be a negative number would make Ax+By a smaller number

Since A and B are relatively prime, this means that GCD(A,B) = 1 and that this divides A+B+AB. So there exists an integer solution for Ax+By = A+B+AB

Hopefully that is enough to prove what you need

Answer 3

well that's even bigger than A+B+AB and it's also bigger than A+B

So you need a bigger positive solution (if it exists), by bigger I mean either x and/or y have to be larger integers

Answer 4

but you know that there's an integer solution since GCD(A,B) = 1, which means it divides C = A+B+AB+k where k is some positive integer

Answer 5

So there's a positive integer solution (x,y) to Ax+By = C where C = A+B+AB+k and k is some positive integer

So this proves the case when C > A +B + AB