灯下 · 几何原本 · 第7卷命题 25

内容第7卷 · 200

命题 Propositio VII.25

If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

若两数互质，则其中一个数自乘所得的积与另一个数互质。

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本页以“自乘数与质数互质”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。

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线

正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。

分步证明Step-by-step proof

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Let A, B be two numbers prime to one another, and let A by multiplying itself make C: I say that B, C are prime to one another. For let D be made equal to A.
设A、B互质，A自乘得C。
Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.
取D等于A，则D与B互质。
Therefore each of the two numbers D, A is prime to B; therefore the product of D, A will also be prime to B.
因此D和A均与B互质，根据VII.24，D与A的积与B互质。
[VII. 24] But the number which is the product of D, A is C.
而D与A的积为C，故C与B互质。

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