Proposition VII.30 — If two numbers by multiplying one another make some number, …

Proof

If a prime

p ∣ ab

but

p ∤ a

, then by VII.29

p

is coprime to

a

; by VII.24 (taking

b

as the multiplicand)

p

must divide

b

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Depends on (2)

VII.24Proposition VII.24If two numbers be prime to any number, their product also will be prime to the same.
VII.29Proposition VII.29Any prime number is prime to any number which it does not measure.

Required by (dependents) (6)

VIII.14Proposition VIII.14If a square measure a square, the side will also measure the side; and if the side measure the side, the square will…
IX.12Proposition IX.12If as many numbers as we please beginning from a unit be in continued proportion, by whatever prime numbers the last is…
IX.13Proposition IX.13If as many numbers as we please beginning from a unit be in continued proportion, and the number after the unit be…
IX.14Proposition IX.14If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except…
IX.16Proposition IX.16If two numbers be prime to one another, the second will not be to any other number as the first is to the second.
IX.32Proposition IX.32Each of the numbers which are continually doubled beginning from a duad is even-times even only.

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