# Diophantus of Alexandria/Arithmetica/Book 1

## Problems by Diophantus of Alexandria: Arithmetica Book $\text I$

### Problem $1$

To divide a given number into two having a given difference.

### 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.