To find the number which is the least that has given parts.

Let *A, B,* and *C* be the given parts.

It is required to find the number which is the least that will have the parts *A, B,* and *C.*

Let *D, E,* and *F* be numbers called by the same name as the parts *A, B,* and *C.* Take *G,* the least number measured by *D, E,* and *F.*

Therefore *G* has parts called by the same name as *D, E,* and *F.*

But *A, B,* and *C* are parts called by the same name as *D, E,* and *F,* therefore *G* has the parts *A, B,* and *C.*

I say next that it is also the least number that has.

If not, there is some number *H* less than *G* which has the parts *A, B,* and *C.*

Since *H* has the parts *A, B,* and *C,* therefore *H* is measured by numbers called by the same name as the parts *A, B,* and *C.* But *D, E,* and *F* are numbers called by the same name as the parts *A, B,* and *C,* therefore *H* is measured by *D, E,* and *F.*

And it is less than *G,* which is impossible. Therefore there is no number less than *G* that has the parts *A, B,* and *C.*

Q.E.D.

Suppose you want to find the smallest number with given parts, say, a fourth part and a sixth part. Then take the LCM(4,6) which is 12. The number 12 has a 1/4 part, namely 3, and a 1/6 part, namely 2.