First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [[VII. 16](elem.7.16)]

Therefore A, B measure C

I say next that it is also the least number they measure.

For, if not, A, B will measure some number which is less than C.

Let them measure D.

Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [[VII. Def. 15](elem.7.def.15)] therefore the product of A, E is equal to the product of B, F.

Therefore, as A is to B, so is F
E. [[VII. 19](elem.7.19)]

But A, B are prime, primes are also least, [[VII. 21](elem.7.21)] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [[VII. 20](elem.7.20)] therefore B measures E, as consequent consequent.

And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [[VII. 17](elem.7.17)]

But B measures E; therefore C also measures D, the greater the less: which is impossible.

Therefore A, B do not measure any number less than C; therefore C is the least that is measured by A, B.

Next, let A, B not be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken; [[VII. 33](elem.7.33)] therefore the product of A, E is equal to the product of B, F. [[VII. 19](elem.7.19)]

And let A by multiplying E make C; therefore also B by multiplying F has made C; therefore A, B measure C.

I say next that it is also the least number that they measure.

For, if not, A, B will measure some number which is less than C.

Let them measure D.

And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H.

Therefore A by multiplying G has made D, and B by multiplying H has made D.

Therefore the product of A, G is equal to the product of B, H; therefore, as A is to B, so is H to G. [[VII. 19](elem.7.19)]

But, as A is to B, so is F to E.

Therefore also, as F is to E, so is H to G.

But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [[VII. 20](elem.7.20)] therefore E measures G.

And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [[VII. 17](elem.7.17)]

But E measures G; therefore C also measures D, the greater the less: which is impossible.

Therefore A, B will not measure any number which is less than C.

Therefore C is the least that is measured by A, B. Q. E. D.