Proposition VII.22 — The least numbers of those which have the same ratio with th…

Proof

Converse of VII.21: any common measure of the least pair would generate a strictly smaller pair in the same ratio, contradicting minimality.

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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.21Proposition VII.21Numbers prime to one another are the least of those which have the same ratio with them.

Required by (dependents) (1)

VII.33Proposition VII.33Given as many numbers as we please, to find the least of those which have the same ratio with them.

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