Bezout jars

Imagine two jars, for example, one with capacity of 7 liters and the other with capacity of 5 liters. The objective is, from this configuration, to obtain exactly 1 liter of water. The motion cycle is:

• Transfers the water
• Empty a jar
• Fills a jar to the top

Does this problem have a solution? Let’s test with the jars below?

What if the jars have different sizes?

Bezout’s Theorem gives an answer to these questions:

Be an equation of the type ax + by = c, with a, b, c integers

The theorem guarantees the existence of infinitely many integer solutions, whenever c is a multiple of the gratest commun divisor of a and b.

In other word, \exist (x,y)\in \mathbb{Z} t.q. ax+by=c whenever c|(a, b).

As a direct consequence of this theorem, whenever a and b are relatively prime, the problem has a solution for any value of c, which is exactly the case with the jars from the example.