---
book: 7
number: 4
id: "VII.4"
kind: "theorem"
uses: ["[[book-7/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.4"
license: "CC-BY-SA-4.0"
---

# VII.4

*Any number is either a part or parts of any number, the less of the greater.*

## Proof

Let *A*, *BC* be two numbers, and let *BC* be the less; I say that *BC* is either a part, or parts, of *A*.

For *A*, *BC* are either prime to one another or not.

First, let *A*, *BC* be prime to one another.

Then, if *BC* be divided into the units in it, each unit of those in *BC* will be some part of *A*; so that *BC* is parts of *A*.

Next let *A*, *BC* not be prime to one another; then *BC* either measures, or does not measure, *A*.

If now *BC* measures *A*, *BC* is a part of *A*.

But, if not, let the greatest common measure *D* of *A*, *BC* be taken; [[book-7/proposition-2|VII. 2]] and let *BC* be divided into the numbers equal to *D*, namely *BE*, *EF*, *FC*.

Now, since *D* measures *A*, *D* is a part of *A*.

But *D* is equal to each of the numbers *BE*, *EF*, *FC*; therefore each of the numbers *BE*, *EF*, *FC* is also a part of *A*; so that *BC* is parts of *A*.

Therefore etc. Q. E. D.