---
book: 9
number: 7
id: "IX.7"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.7"
license: "CC-BY-SA-4.0"
---

# IX.7

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

## Proof

For let the composite number *A* by multiplying any number *B* make *C*; I say that *C* is solid.

For, since *A* is composite, it will be measured by some number. [[book-7/definitions#Definition 13|VII. Def. 13]]

Let it be measured by *D*; and, as many times as *D* measures *A*, so many units let there be in *E*.

Since then *D* measures *A* according to the units in *E*, therefore *E* by multiplying *D* has made *A*. [[book-7/definitions#Definition 15|VII. Def. 15]]

And, since *A* by multiplying *B* has made *C*, and *A* is the product of *D*, *E*, therefore the product of *D*, *E* by multiplying *B* has made *C*.

Therefore *C* is solid, and *D*, *E*, *B* are its sides. Q. E. D.