I answered this question incorrectly. I understood my error after reading Bunuel solution.

Please allow me to provide an alternative explanation of the second part. Which is easier for me to understand.

...Alternative explanation to Bunuel's solution of part 2...(2) 5a−b=1. Rearrange: 5a=b+1.

Still using a=md and b=nd. We have 5(md)-(nd)=d(5d-n)=1. Since d and (5d-n) need to be positive Integers, the only possible solution to the equation is that both are equal to 1. Therefore d=1 and GCD(x,y)=d=1

Bunuel. Many Thanks for your solution