第7卷命题 36 · 求三数的最小公倍数

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.

取A、B所度量的最小数D。若C度量D，则A、B、C共同度量D。

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.

假设A、B、C度量某个小于D的数E，则A、B度量E，故D度量E，大度量小，矛盾。因此D是所求最小数。

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.

若C不度量D，取C、D所度量的最小数E。因A、B度量D且D度量E，故A、B度量E；又C度量E，故A、B、C度量E。

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.

假设A、B、C度量某个小于E的数F，则A、B度量F，故D度量F；又C度量F，故D、C度量F，从而E度量F，大度量小，矛盾。因此E是所求最小数。