Proposition VII.21 — Numbers prime to one another are the least of those which ha…

Proof

If two coprime numbers

a

b

had smaller numbers

c

d

with

a : b = c : d

, then by VII.20

c

d

would measure

a

b

a common number of times; the common measure would contradict coprimality.

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Depends on (2)

VII.20Proposition VII.20The least numbers of those which have the same ratio with them measure those which have the same ratio the same number…
VII.12Definition VII.12Numbers prime to one another are those which are measured by a unit alone as a common measure.

Required by (dependents) (3)

VII.22Proposition VII.22The least numbers of those which have the same ratio with them are prime to one another.
VIII.1Proposition VIII.1If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the…
VIII.3Proposition VIII.3If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the…

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