Proposition VII.4 — Any number is either a part or parts of any number, the less…

Proof

Either the less divides the greater (a part, Definition VII.3) or it does not (parts, Definition VII.4). The Euclidean algorithm terminates, so the case analysis is exhaustive.

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Depends on (3)

VII.2Proposition VII.2Given two numbers not prime to one another, to find their greatest common measure.
VII.3Definition VII.3A number is a part of a number, the less of the greater, when it measures the greater.
VII.4Definition VII.4But parts when it does not measure it.

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