---
book: 7
number: 3
id: "VII.3"
kind: "construction"
uses: ["[[book-7/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.3"
license: "CC-BY-SA-4.0"
---

# VII.3

*Given three numbers not prime to one another, to find their greatest common measure.*

## Proof

Let *A*, *B*, *C* be the three given numbers not prime to one another; thus it is required to find the greatest common measure of *A*, *B*, *C*.

For let the greatest common measure, *D*, of the two numbers *A*, *B* be taken; [[book-7/proposition-2|VII. 2]] then *D* either measures, or does not measure, *C*.

First, let it measure it.

But it measures *A*, *B* also; therefore *D* measures *A*, *B*, *C*; therefore *D* is a common measure of *A*, *B*, *C*.

I say that it is also the greatest.

For, if *D* is not the greatest common measure of *A*, *B*, *C*, some number which is greater than *D* will measure the numbers *A*, *B*, *C*.

Let such a number measure them, and let it be *E*.

Since then *E* measures *A*, *B*, *C*, it will also measure *A*, *B*; therefore it will also measure the greatest common measure of *A*, *B*. [[book-7/proposition-2|VII. 2, Por.]]

But the greatest common measure of *A*, *B* is *D*; therefore *E* measures *D*, the greater the less: which is impossible.

Therefore no number which is greater than *D* will measure the numbers *A*, *B*, *C*; therefore *D* is the greatest common measure of *A*, *B*, *C*.

Next, let *D* not measure *C*; I say first that *C*, *D* are not prime to one another.

For, since *A*, *B*, *C* are not prime to one another, some number will measure them.

Now that which measures *A*, *B*, *C* will also measure *A*, *B*, and will measure *D*, the greatest common measure of *A*, *B*. [[book-7/proposition-2|VII. 2, Por.]]

But it measures *C* also; therefore some number will measure the numbers *D*, *C*; therefore *D*, *C* are not prime to one another.

Let then their greatest common measure *E* be taken. [[book-7/proposition-2|VII. 2]]

Then, since *E* measures *D*, and *D* measures *A*, *B*, therefore *E* also measures *A*, *B*.

But it measures *C* also; therefore *E* measures *A*, *B*, *C*; therefore *E* is a common measure of *A*, *B*, *C*.

I say next that it is also the greatest.

For, if *E* is not the greatest common measure of *A*, *B*, *C*, some number which is greater than *E* will measure the numbers *A*, *B*, *C*.

Let such a number measure them, and let it be *F*.

Now, since *F* measures *A*, *B*, *C*, it also measures *A*, *B*; therefore it will also measure the greatest common measure of *A*, *B*. [[book-7/proposition-2|VII. 2, Por.]]

But the greatest common measure of *A*, *B* is *D*; therefore *F* measures *D*.

And it measures *C* also; therefore *F* measures *D*, *C*; therefore it will also measure the greatest common measure of *D*, *C*. [[book-7/proposition-2|VII. 2, Por.]]

But the greatest common measure of *D*, *C* is *E*; therefore *F* measures *E*, the greater the less: which is impossible.

Therefore no number which is greater than *E* will measure the numbers *A*, *B*, *C*; therefore *E* is the greatest common measure of *A*, *B*, *C*. Q. E. D.