Propositiones ad Acuendos Juvenes/Problems/39 - De Quodam Emptore in Oriente

by : Problem $39$

 * De Quodam Emptore in Oriente: An Oriental Merchant
 * A man in the east wanted to buy $100$ assorted animals for $100$ shillings.
 * He ordered his servant to pay $5$ shillings for a camel,
 * $1$ shilling for a donkey,
 * ''and $1$ shilling for $20$ sheep.


 * How many camels, donkeys and sheep did he buy?

Solution

 * $19$ camels, $1$ donkey and $80$ sheep.

Proof
Let $c$, $d$ and $s$ denote the number of camels, donkeys and sheep respectively.

We have:

Note that both $c$ and $d$ need to be (strictly) positive.

We need to find possible values of $c$ such that $2300 - 71 c$ is divisible by $19$.

This can happen only when $c$ itself is divisible by $19$.

It is implicit that there are at least some camels are bought, so the solution:
 * $c = 0, d = 100, s = 0$

is usually ruled out.

Hence we have:
 * $c = 19, d = 1, s = 80$