# Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm

### (Proposition 2 from Book 7 of Euclid's “Elements”)

To find the greatest common measure of two given numbers (which are) not prime to one another. * Let $AB$ and $CD$ be the two given numbers (which are) not prime to one another. * So it is required to find the greatest common measure of $AB$ and $CD$.

### Notes

• Euclid does not provide a definition of the greatest common divisor, he only provides a method to calculate it.
• 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. There exists also a correct and efficient algorithm for finding the $$\gcd$$ of any two integers, at least one of which does not equal $$0$$.
• The contemporary definition of the greatest common divisor not given in Euclid's Elements has the advantage that $$gcd(0,0)$$ is well-defined: $$0$$ is the only integer, which is divisible by all other integers. Therefore, $$\gcd(0,0)=0$$.

Proofs: 1 Corollaries: 1

Proofs: 1 2

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