To find the greatest common measure of three given numbers
not relatively prime.

Let *A, B,* and *C* be the three given numbers not relatively prime.

It is required to find the greatest common measure of *A, B,* and *C.*

Take the greatest common measure, *D,* of the two numbers *A* and *B.* Then either *D* measures, or does not measure, *C.*

First, let *D* measure *C.*

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

I say that it is also the greatest.

If *D* is not the greatest common measure of *A, B,* and *C,* then some number *E,* greater than *D,* measures the numbers *A, B,* and *C.*

Since then *E* measures *A, B,* and *C,* therefore it measures *A* and *B.* Therefore it also measures the greatest common measure of *A* and *B.* But the greatest common measure of *A* and *B* is *D,* therefore *E* measures *D,* the greater the less, which is impossible.

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

Next, let *D* not measure *C.*

I say first that *C* and *D* are not relatively prime.

Since *A, B,* and *C* are not relatively prime, therefore some number measures them.

Now that which measures *A, B,* and *C* also measures *A* and *B,* and therefore measures *D,* the greatest common measure of *A* and *B.* But it measures *C* also, therefore some number measures the numbers *D* and *C.* Therefore *D* and *C* are not relatively prime.

Take their greatest common measure *E.*

Then, since *E* measures *D,* and *D* measures *A* and *B,* therefore *E* also measures *A* and *B.* But it measures *C* also, therefore *E* measures *A, B,* and *C.* Therefore *E* is a common measure of *A, B,* and *C.*

I say next that it is also the greatest.

If *E* is not the greatest common measure of *A, B,* and *C,* then some number *F,* greater than *E,* measures the numbers *A, B,* and *C.*

Now, since *F* measures *A, B,* and *C,* it also measures *A* and *B,* therefore it measures the greatest common measure of *A* and *B.* But the greatest common measure of *A* and *B* is *D,* therefore *F* measures *D.*

And it measures *C* also, therefore *F* measures *D* and *C.* Therefore it also measures the greatest common measure of *D* and *C.* But the greatest common measure of *D* and *C* is *E,* therefore *F* measures *E,* the greater the less, which is impossible.

Therefore no number which is greater than *E* measures the numbers *A, B,* and *C.* Therefore *E* is the greatest common measure of *A, B,* and *C.*

Q.E.D.

This proposition constructs the GCD(*a, b, c*) as
GCD(GCD(*a, b*), *c*).

The proof that this construction works is simplified if 1 is considered to be a number. Then, two numbers are relatively prime when their GCD is 1, and Euclid’s first case in the proof is subsumed in the second.

Let *d* = GCD(*a, b*), and let
*e* = GCD(*d, c*). Since
*e* | *d, d* | *a,* and
*d* | *b,* it follows that *e* | *a*
and *e* | *b,* so *e,* in fact, is a common divisor
of *a, b,* and *c.*

In order to show that *e* is the greatest common divisor, let *f* be any common divisor
of *a, b,* and *c.* Then as *f* | *a* and
*f* | *b,* therefore *f* | GCD(*a, b*),
that is, *f* | *d.* Also, as *f* | *d* and
*f* | *c,* therefore *f* | GCD(*d, c*),
that is *f* | *e.* Therefore *e* is the greatest common divisor
of *a, b,* and *c.* Q.E.D.

This is the same proposition as X.4.

This proposition is used in VII.33.