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G.276 least numbers ratio¶

The least numbers of those which have the same ratio with them are prime to one another.

least_numbers_ratio_vii_22(A, B)
given A, B be the least numbers of those which have the same ratio with them
prove A, B are prime to one another

given A, B be the least numbers of those which have the same ratio with them

assert If they are not prime to one another, some number will measure them

assert Let some number measure them, and let it be C

assert as many times as C measures A, so many units let there be in D, and, as many times as C measures B, so many units let there be in E

assert C measures A according to the units in D,

∵ number_said_multiply_defassert C by multiplying D has made A

assert For the same reason also

assert C by multiplying E has made B

assert the number C by multiplying the two numbers D, E has made A, B

∵ number_multiplying_numbersassert as D is to E, so is A to B

assert D, E are in the same ratio with A, B, being less than they ⇒⇐

assert no number will measure the numbers A, B

∴ A, B are prime to one another

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