Proof
Suppose the extremes shared a common divisor . By VII.20
each term would be divisible by some power of , producing a
smaller sequence in the same ratio — contradicting minimality.
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Depends on (3)
- 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.
- 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…
Required by (dependents) (3)
- VIII.7Proposition VIII.7If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the…
- IX.15Proposition IX.15If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever…
- IX.17Proposition IX.17If as many numbers as we please be in continued proportion, and the extremes of them be prime to one another, the last…
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