## Proposition 39

 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. VII.36 Therefore G has parts called by the same name as D, E, and F. VII.37 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. VII.38 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.
The wording of the proposition is somewhat unclear, but an example will show its intent.

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.

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