VII.34

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

Proof

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.