Greek Anthology Book XIV: 48. - Problem

Problem

 * The Graces were carrying baskets of apples, and in each was the same number.
 * The nine Muses met them and asked them for apples, and they gave the same number to each Muse,
 * and the nine and three had each of them the same number.


 * Tell me how many they gave and how they all had the same number.

Solution
Let $n$ be the number of apples owned by each party at the end of the transaction.

At the end of the transaction, there are $12$ parties each with the same number $n$.

Each of the Graces then had $4 n$ apples at the start.

In total there were $12 n$ apples.

Any (strictly) positive integer $n$ satisfies the conditions.