[issue22477] GCD in Fractions

[Mark Dickinson]

Mark Dickinson added the comment:

> The negative of the greatest common divisor is the least common divisor in an integer range.

That depends on your choice of definitions: it's perfectly reasonable to see it as another greatest common divisor, if you interpret "greatest" as being with respect to the divisibility lattice, not the total ordering of Z.  That's in some sense the correct interpretation, because it's the one that generalises to other interesting rings: for example, the Gaussian integers have a well-defined and useful notion of greatest common divisor, but aren't ordered, and the ring Z[sqrt 3] similarly has well-defined greatest common divisors (defined up to multiplication by a unit, as usual) *and* a total ordering, but "greatest" *can't* be interpreted in the ordering sense in that case (because there are infinitely many units).

Many textbooks will talk about "a greatest common divisor" rather than "the greatest common divisor".  In that sense, -3 *is* a greatest common divisor of 6 and -15.

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<http://bugs.python.org/issue22477>
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