The ass and mule problem is a classic arithmetical problem preserved in a Greek epigram addressed to Euclid (Leventhal 2022, pp. 192-193). It is formulated as a dialogue
between the two animals after which it is named. The mule says 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 
of sacks of the mule and the number 
of sacks of the ass are related by the
identities:
 |  |  |
(1)
|
 |  |  |
(2)
|
which form a system of two linear equations. The solution is 
, 
.
A generalization of this problem has been studied by Singmaster (1999, 2002). In this modified problem, the ass says, "If you gave me 
of your sacks, I would have 
times as many as you," and the mule answers, "If
you gave my 
of your sacks, I would have 
times as many as you." This has solutions
 |  |  |
(3)
|
 |  |  |
(4)
|
which are integers only when 
and 
.
