The Merchants and the Coin Purse

Three merchants saw in the road a purse [containing money].
One said, "If I secure this purse, I shall become twice as rich as both of you together."
Then the second said, "I shall become three times as rich."
Then the third said, "I shall become five times as rich."
What is the value of the money in the purse, as also the money on hand [with each of the three merchants]?

(Source: The Penguin Book of Curious and Interesting Puzzles by David Wells, this puzzle by way of the Bakhshali manuscript)

Let $p$ be the value of the contents of the purse and $m_1$, $m_2$ and $m_3$ be the values of the cash on hand held by the first, second and third merchants, respectively. If you read the problem right, this problem becomes a system of linear equations:

\begin{align*}
p + m_1 &= 2(m_2 + m_3) \\
p + m_2 &= 3(m_1 + m_3) \\
p + m_3 &= 5(m_2 + m_3)
\end{align*}

You may notice that there are four unknowns and only three equations, making the system underdetermined. (But that's just as well given that my knowledge of medieval Indian currency is effectively nonexistent.) That being said, to the extent that it can be solved—I did so in Maxima—the answer is $p = 3$, $m_1 = \frac{r_1}{5}$, $m_2 = \frac{3 r_1}{5}$, $m_3 = r_1$, where $r_1$ is effectively an arbitrary constant. But if we hone in on the solution that has the smallest values possible without allowing fractional coins, then the merchants, in order, hold one, three and five coins of equal value, respectively, and the purse contains 15 coins of that value. This means, for example, that by securing the purse, the first merchant would then hold 16 coins, twice as many as three held by the second and the five held by the third added together.