Proposition VII.24 — If two numbers be prime to any number, their product also wi…

Proof

g cd (a, c) = 1

and

g cd (b, c) = 1

, then any prime dividing

ab

and

c

must divide

a

b

(using the descent from VII.23) and would contradict one of the hypotheses.

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Depends on (1)

VII.23Proposition VII.23If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.

Required by (dependents) (5)

VII.25Proposition VII.25If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
VII.26Proposition VII.26If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
VII.30Proposition VII.30If two numbers by multiplying one another make some number, and any prime number measure the product, it will also…
IX.30Proposition IX.30If an odd number measure an even number, it will also measure the half of it.
IX.31Proposition IX.31If an odd number be prime to any number, it will also be prime to the double of it.

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