If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.
若两数互质，则其和与其中每一数也互质；反之，若两数之和与其中一数互质，则原两数也互质。
本页以“互质数与和互质定理”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For let two numbers AB, BC prime to one another be added; I say that the sum AC is also prime to each of the numbers AB, BC. For, if CA, AB are not prime to one another, some number will measure CA, AB. Let a number measure them, and let it be D. Since then D measures CA, AB, therefore it will also measure the remainder BC.
设两数AB、BC互质，其和为AC。假设AC与AB不互质，则存在数D同时度量AC和AB。
But it also measures BA; therefore D measures AB, BC which are prime to one another: which is impossible. [VII. Def. 12] Therefore no number will measure the numbers CA, AB; therefore CA, AB are prime to one another. For the same reason AC, CB are also prime to one another.
由于D度量AC和AB，则D也度量差BC；但D也度量AB，故D度量互质的AB和BC，矛盾。
Therefore CA is prime to each of the numbers AB, BC. Again, let CA, AB be prime to one another; I say that AB, BC are also prime to one another. For, if AB, BC are not prime to one another, some number will measure AB, BC. Let a number measure them, and let it be D.
因此AC与AB互质，同理AC与BC互质。
Now, since D measures each of the numbers AB, BC, it will also measure the whole CA. But it also measures AB; therefore D measures CA, AB which are prime to one another: which is impossible. [VII. Def. 12] Therefore no number will measure the numbers AB, BC.
反之，设AC与AB互质。假设AB与BC不互质，则存在数D度量AB和BC，从而D也度量和AC；但D也度量AB，故D度量互质的AC和AB，矛盾。因此AB与BC互质。