First, concerning a).

You need to show that there

*exists* an element in

, not that

consists of positive elements or something like that. When you prove existence,

*you* have the right to choose an element. In this case, you can choose

, which belongs to

with

,

. This is different from when you are proving something

*for all* and

: then it is your opponent (so to speak; the one who doubts the validity of the statement) and not you who has the right to choose

and

.

You have already shown that it exists: it is

. Indeed, it is a common divisor by d), and every other common divisor

divides

by c), so every other common divisor

does not exceed

.

This is not correct; c) shows that only the smallest element

of

divides

and

. You used the fact that

is the smallest when you proved that the remainder of

is 0, since it is smaller than

. Also, e.g.,

, but

does not have to divide

.

You need to show not that the least element of

is unique but that the GCD is unique. Not every common divisor of

and

is in

; for example, when

and

are both even, then every element of

is even and therefore

.

But the idea is basically correct. If there are two GCDs, then each of them is

than the other, so they are equal.