Product of Composite Number with Number is Solid Number

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


Sources