| Any number is either a part or parts of any number, the less of the greater. | ||
| Let A and BC be two numbers, and let BC be the less.
I say that BC is either a part, or parts, of A. | ||
| Either A and BC are relatively prime or they are not.
First, let A and BC be relatively prime. | ||
| Then, if BC is divided into the units in it, then each unit of those in BC is some part of A, so that BC is parts of A. | VII.Def.4 | |
| Next let A and BC not be relatively prime, then BC either measures, or does not measure, A. | ||
| Now if BC measures A, then BC is a part of A. But, if not, take the greatest common measure D of A and BC, and divide BC into the numbers equal to D, namely BE, EF, and FC. | VII.Def.3
VII.2 | |
| Now, since D measures A, therefore D is a part of A. But D equals each of the numbers BE, EF, and FC, therefore each of the numbers BE, EF, and FC is also a part of A, so that BC is parts of A. | ||
| Therefore, any number is either a part or parts of any number, the less of the greater. | ||
| Q.E.D. | ||
The proof is simplified if 1 is considered to be a number. Let a and b be two numbers with a > b. Using VII.2, let d = GCD(a, b). Then d divides both a and b, that is, d is a part of a and a part of b. Let e be the number of times that d measures b, that is, e = b/d. Then b is e parts of a. (If e = 1, then b is just one part of a.) Q.E.D.
This proposition is used in VII.20.
Book VII Introduction - Proposition VII.3 - Proposition VII.5.