previous next


If two numbers be prime to any number, their product also will be prime to the same.

For let the two numbers A, B be prime to any number C, and let A by multiplying B make D; I say that C, D are prime to one another.

For, if C, D are not prime to one another, some number will measure C, D.

Let a number measure them, and let it be E.

Now, since C, A are prime to one another, and a certain number E measures C, therefore A, E are prime to one another. [VII. 23]

As many times, then, as E measures D, so many units let there be in F; therefore F also measures D according to the units in E. [VII. 16]

Therefore E by multiplying F has made D. [VII. Def. 15]

But, further, A by multiplying B has also made D; therefore the product of E, F is equal to the product of A, B.

But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [VII. 19] therefore, as E is to A, so is B to F.

But A, E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [VII. 21] and the least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures B.

But it also measures C; therefore E measures B, C which are prime to one another: which is impossible. [VII. Def. 12]

Therefore no number will measure the numbers C, D.

Therefore C, D are prime to one another. Q. E. D. 1

1 ἐξ αὐτῶν γενόμενος, literally “the (number) produced from them,” will henceforth be translated as “their product.”

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.

load focus Greek (J. L. Heiberg, 1883)
hide Display Preferences
Greek Display:
Arabic Display:
View by Default:
Browse Bar: