Problems with buckets of water(discussion)

Let's say we have 3 unscaled buckets A,B,C of capacity 8,5 and 20 l respectively,and C is full of water.Measure 6l of water,where the only operation allowed is the spilling from one bucket to another until one is full or the other is empty.

What if we had A(8l),B(6l),C(20l) could we measure 7l of water with the same operation ?

In first case the answer is yes,the second case the answer is no:

well let's define (x,y,z) the quantity of water in (A,B,C);

Initially we had (0,0,20);

Then $ (0,0,20) \rightarrow (8,0,12) \rightarrow (3,5,12) \rightarrow (3,0,17) \rightarrow (0,3,17) \rightarrow (8,3,9) \rightarrow(6,5,9)$

More generally if we have three buckets with capacity of a , b,c respectively and another bucket large enough to transfer the water if necessary,we can measure e litres of water if and only if $latex ax+by=e$has a pair of integers as solution.

So,if $ (a,b,c)=d$ where $(a,b,c)$ means the greatest common divisor of a, b and c and $ d\nmid e$ then we cannot measure with the three buckets .So we need to introduce the fourth bucket with capacity f where $ (a,b,c,f)=1$.

There should be at least two proofs for this,and I expect them from you.