I will comment on part (a) for now, and then when I have more time, I will comment on the remaining parts (unless someone else beats me to it).
Your thinking is correct. Suppose that a and b have a common factor x>1. Then a = mx and b=nx for some integers m and n. But then:
ad-bc=mxd-nxc = x(md-nc). Here both x and md-nc are integers, and furthermore x>1. Hence their product cannot possibly be 1.
(Why? If md-nc>0, then x(md-nc) is greater than or equal to x, which is strictly greater than 1. If md-nc=0, then the product is 0. If md-nc<0, then x(md-nc) < md-nc <0, which thus can't be 1)