A classic arithmetical problem probably first posed by Euclid and investigated by various authors in the Middle Ages.
The problem is formulated as a dialogue between the two animals after which it is called. The mule say to the ass, "If you gave me one of your sacks, I would have as many as you." The ass replies, "If you gave one of your sacks, I would have twice as many as you." The question, of course, is "How many sacks do they have?" The number x of sacks of the mule and the number y of sacks of the ass are related by the identities:


which form a system of two linear equations. The solution is x = 5, y = 7.

A generalization of this problem has been studied by Singmaster (1999, 2002). In this modified problem, the ass says, "If you gave my a of your sacks, I would have b times as many as you," and the mule answers, "If you gave my c of your sacks, I would have d times as many as you." This has solutions


which are integers only when and