| 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. | |
Book VII Introduction - Proposition VII.34 - Proposition VII.36.