# Definition:Solid Number

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## Definition

A **solid number** is the product of three (natural) numbers.

In the words of Euclid:

*And, when three numbers having multiplied one another make some number, the number so produced is***solid**, and its**sides**are the numbers which have multiplied one another.

(*The Elements*: Book $\text{VII}$: Definition $17$)

### Side

The **side** of a solid number is one of the (natural) numbers which are its divisors.

### Example

A example of a solid number is $12 = 2 \times 2 \times 3$.

Its divisors $2$ and $3$ are its sides.

## Similar Numbers

Let $m$ and $n$ be solid numbers.

Let:

- $m = p_1 \times p_2 \times p_3$ where $p_1 \le p_2 \le p_3$
- $n = q_1 \times q_2 \times q_3$ where $q_1 \le q_2 \le q_3$

Then $m$ and $n$ are **similar** iff:

- $p_1 : q_1 = p_2 : q_2 = p_3 : q_3$

## Also see

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$