---
book: 7
number: 34
id: "VII.34"
kind: "construction"
uses: ["[[book-7/proposition-16]]", "[[book-7/proposition-19]]", "[[book-7/proposition-21]]", "[[book-7/proposition-20]]", "[[book-7/proposition-17]]", "[[book-7/proposition-33]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.34"
license: "CC-BY-SA-4.0"
---

# 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*. [[book-7/proposition-16|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*; [[book-7/definitions#Definition 15|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*. [[book-7/proposition-19|VII. 19]]

But *A*, *B* are prime, primes are also least, [[book-7/proposition-21|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; [[book-7/proposition-20|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*. [[book-7/proposition-17|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; [[book-7/proposition-33|VII. 33]] therefore the product of *A*, *E* is equal to the product of *B*, *F*. [[book-7/proposition-19|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*. [[book-7/proposition-19|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; [[book-7/proposition-20|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*. [[book-7/proposition-17|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.