The least numbers of those which have the same ratio with them are prime to one another.
与它们有相同比的最小两数互素。
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Let A, B be the least numbers of those which have the same ratio with them; I say that A, B are prime to one another. For, if they are not prime to one another, some number will measure them.
设A、B是与它们有相同比的最小两数。
Let some number measure them, and let it be C. And, as many times as C measures A, so many units let there be in D, and, as many times as C measures B, so many units let there be in E Since C measures A according to the units in D, therefore C by multiplying D has made A.
假设A、B不互素，则存在某数C同时度量A和B。
[VII. Def. 15] For the same reason also C by multiplying E has made B. Thus the number C by multiplying the two numbers D, E has made A, B; therefore, as D is to E, so is A to B; [VII. 17] therefore D, E are in the same ratio with A, B, being less than they: which is impossible.
设C度量A得D个单位，度量B得E个单位，则C乘以D得A，C乘以E得B。
Therefore no number will measure the numbers A, B.
因此D比E等于A比B，且D、E小于A、B，与A、B最小性矛盾，故A、B互素。