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Proposition: 7.03: Greatest Common Divisor of Three Numbers

(Proposition 3 from Book 7 of Euclid's “Elements”)

To find the greatest common measure of three given numbers (which are) not prime to one another. * Let $A$, $B$, and $C$ be the three given numbers (which are) not prime to one another. * So it is required to find the greatest common measure of $A$, $B$, and $C$.

Modern Formulation

See the greatest common divisor of more than two numbers.

Notes

• Nowadays, it is not necessary to require that the numbers are not prime to one another to be able to find their greatest common divisor.
• Euclid does not provide a definition of the greatest common divisor, he only provides a method to calculate it.
• The modern definition is a generalization of the greatest common divisor of two numbers for any finite number of integers, not only of three integers.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1

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References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
2. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Bibliography

1. Health L. Thomas (Transl.): "Euclid's Elements - all thirteen books", Green Lion Press, 2013,