If two numbers measure any number, then the least number measured by them also measures the same.

Let the two numbers *A* and *B* measure any number *CD,* and let *E* be the least that they measure.

I say that *E* also measures *CD.*

If *E* does not measure *CD,* let *E,* measuring *DF,* leave *CF* less than itself.

Now, since *A* and *B* measure *E,* and *E* measures *DF,* therefore *A* and *B* also measure *DF.* But they also measure the whole *CD,* therefore they measure the remainder *CF* which is less than *E,* which is impossible.

Therefore *E* cannot fail to measure *CD.* Therefore it measures it.

Therefore, *if two numbers measure any number, then the least number measured by them also measures the same.*

Q.E.D.