According to the explaination if gcd(A,B) != 1 then next possible value of X is A/gcd(A,B).

Why they have ruled out the next highest factor of A as a possibility.

For ex let the next highest factor of A after A be Z. Then gcd(Z,B) can also be 1.

Why they have ruled out the next highest factor of A as a possibility.

For ex let the next highest factor of A after A be Z. Then gcd(Z,B) can also be 1.