To find the least number which three given numbers measure.

Let *A, B,* and *C* be the three given numbers.

It is required to find the least number which they measure.

Take *D* the least number measured by the two numbers *A* and *B.*

Then *C* either measures, or does not measure, *D.*

First, let it measure it.

But *A* and *B* also measure *D,* therefore *A, B,* and *C* measure *D.*

I say next that it is also the least that they measure.

If not, *A, B,* and *C* measure some number *E* less than *D.*

Since *A, B,* and *C* measure *E,* therefore *A* and *B* measure *E.* Therefore the least number measured by *A* and *B* also measures *E.*

But *D* is the least number measured by *A* and *B,* therefore *D* measures *E,* the greater the less, which is impossible.

Therefore *A, B,* and *C* do not measure any number less than *D.* Therefore *D* is the least that *A, B,* and *C* measure.

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

Take *E,* the least number measured by *C* and *D.*

Since *A* and *B* measure *D,* and *D* measures *E,* therefore *A* and *B* also measure *E.* But *C* also measures *E,* therefore *A, B,* and *C* also measure *E.*

I say next that it is also the least that they measure.

If not, *A, B,* and *C* measure some number *F* less than *E.*

Since *A, B,* and *C* measure *F,* therefore *A* and *B* measure *F.* Therefore the least number measured by *A* and *B* also measures *F.* But *D* is the least number measured by *A* and *B,* therefore *D* measures *F.* But *C* also measures *F,* therefore *D* and *C* measure *F,* so that the least number measured by *D* and *C* also measures *F.*

But *E* is the least number measured by *C* and *D,* therefore *E* measures *F,* the greater the less, which is impossible.

Therefore *A, B,* and *C* do not measure any number which is less than *E.* Therefore *E* is the least that is measured by *A, B,* and *C.*

Q.E.D.

This proposition is used in the proof of proposition VII.39.