If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.
若两数互质，则度量其中一数的数必与另一数互质。
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Let A, B be two numbers prime to one another, and let any number C measure A; I say that C, B are also prime to one another. For, if C, B are not prime to one another, some number will measure C, B.
设A、B两数互质，且C度量A。
Let a number measure them, and let it be D.
假设C与B不互质，则存在某数D度量C与B。
Since D measures C, and C measures A, therefore D also measures A. But it also measures B; therefore D measures A, B which are prime to one another: which is impossible.
因D度量C，且C度量A，故D亦度量A；又D度量B，故D度量互质的A与B。
[VII. Def. 12] Therefore no number will measure the numbers C, B.
此与互质定义矛盾，故无数量度C与B，即C与B互质。