VII.34

Given two numbers, to find the least number which they measure.

Let A, B be the two given numbers; thus it is required to find the least number which they measure.

Now A, B are either prime to one another or not.

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]

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] 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]

But A, B are prime, primes are also least, [VII.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] 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]

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] therefore the product of A, E is equal to the product of B, F. [VII.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]

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] 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]

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.