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Note that by Result 1 our formula for GCRD does not require rational fractions to be in reduced form.
It is not difficult to show that (exercise!): GCRD (a/bg, c/dg) = 1 if g = GCRD (a/b, c/d).
In other words, when two rational numbers are divided by their GCRD, the resulting numbers are relatively prime integers.
Note that the Cranberry Juice problem affords an opportunity for considering GCRD applied to three (or more) terms.
The entire development of GCRD can be repeated with minimal effort to give birth to its twin LCRM, or least common rational multiple.
An unexpected application of GCRD is another proof of the irrationality of [square root of (2)].
Consequently, by Result 2, these algorithms can also be used to calculate GCRD. An alternative definition for GCRD can be formulated by means of factorisation and use of non-positive integer exponents.
(b) GCRD ([e/f] x [a/b], [e/f] x [c/d]) = [e/f] x GCRD (a/b, c/d)
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- GCRC Advisory Committee