Product of Composite Number with Number is Solid Number

Theorem

Let $a, b \in \Z$ be positive integers.

Let $a$ be a composite number.

Then $a b$ is a solid number.

In the words of Euclid:

If a composite number by multiplying any number make some number, the product will be solid.

(The Elements: Book $\text{IX}$: Proposition $7$)

Proof

By definition of composite number:

$\exists p, q \in \Z_{>1}: a = p q$

Then:

$a b = p q b$

Hence the result by definition of solid number.

$\blacksquare$

Historical Note

This proof is Proposition $7$ of Book $\text{IX}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions