# Diophantus of Alexandria/Arithmetica

## Book $\text I$

### Problem $8$

What number must be added to $100$ and to $20$ (the same number added to each) so that the sums are in the ratio $3 : 1$?

### Problem $15$

Two numbers are such that:
if the first receives $30$ from the second, they are in the ratio $2 : 1$
if the second receives $50$ from the first, they are in the ratio $1 : 3$.

What are these numbers?

### Problem $17$

The sums of $4$ numbers, omitting each of the numbers in turn, are $22$, $24$, $27$ and $20$ respectively

What are the numbers?

### Problem $22$

To find $3$ numbers such that, if each give to the next following a given fraction of itself, in order, the results after each has given and taken may be equal.

That is:

Let $\dfrac 1 p, \dfrac 1 q, \dfrac 1 r$ be given.

The exercise is to find a set of $3$ natural numbers $\set {x, y, z}$ such that:

 $\ds x - \frac x p + \frac z r$ $=$ $\ds m$ $\ds y - \frac y q + \frac x p$ $=$ $\ds m$ $\ds z - \frac z r + \frac y q$ $=$ $\ds m$

where $m$ is a natural number.

## Book $\text {III}$

### Problem $6$

To find $3$ numbers such that their sum is a square and the sum of any pair of them is a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p + q + r$ is square
$p + q$ is square
$q + r$ is square
$r + p$ is square.

What are those $3$ numbers?

### Problem $12$

To find $3$ numbers such that the product of any $2$ of them added to the $3$rd gives a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p q + r$ is square
$q r + p$ is square
$r p + q$ is square

What are those $3$ numbers?

## Book $\text V$

### Problem $30$

A man buys a certain number of measures of wine, some at $8$ drachmas, some at $5$ drachmas each.
He pays for them a square number of drachmas;
and if we add $60$ to this number, the result is a square, the side of which is equal to the whole number of measures.
Find how many he bought at each price.