Given three numbers, to find the least number which they measure.

Let A, B, C be the three given numbers; thus it is required to find the least number which they measure.

Let D, the least number measured by the two numbers A, B, be taken. [VII. 34]

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

First, let it measure it.

But A, B also measure D; therefore A, B, C measure D.

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

For, if not, A, B, C will measure some number which is less than D.

Let them measure E.

Since A, B, C measure E, therefore also A, B measure E.

Therefore the least number measured by A, B will also measure E. [VII. 35]

But D is the least number measured by A, B; therefore D will measure E, the greater the less: which is impossible.

Therefore A, B, C will not measure any number which is less than D; therefore D is the least that A, B, C measure.

Again, let C not measure D, and let E, the least number measured by C, D, be taken. [VII. 34]

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

But C also measures E; therefore also A, B, C measure E.

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

For, if not, A, B, C will measure some number which is less than E.

Let them measure F.

Since A, B, C measure F, therefore also A, B measure F; therefore the least number measured by A, B will also measure F. [VII. 35]

But D is the least number measured by A, B; therefore D measures F.

But C also measures F; therefore D, C measure F, so that the least number measured by D, C will also measure F.

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

Therefore A, B, C will not measure any number which is less than E.

Therefore E is the least that is measured by A, B, C. Q. E. D.