Farmer
John has received an order for exactly M units of milk (1≤M≤200) that
he needs to fill right away. Unfortunately, his fancy milking machine
has just become broken, and all he has are two milk pails of integer
sizes X and Y (1≤X,Y≤100) with which he can measure milk. Both pails are
initially empty. Using these two pails, he can perform up to K of the
following types of operations (1≤K≤100):
- He can fill either pail completely to the top.
- He can empty either pail.
- He can pour the contents of one pail
into the other, stopping when the former becomes empty or the latter
becomes full (whichever of these happens first).
Although FJ realizes he may not be able to end up with exactly M total
units of milk in the two pails, please help him compute the minimum
amount of error between M and the total amount of milk in the two pails.
That is, please compute the minimum value of |M−M′| such that FJ can
construct M′ units of milk collectively between the two pails.