the less. Proposition 21 Numbers prime to one another are the least of those that have the same ratio with them. Proposition 29 Any prime number is prime to any number it does not measure. Proposition 30 If two numbers, by multiplying one another, make the same number, and any prime number measures the product, it also measures one of the original numbers. Proof of 30 If c, a prime number, measure ab, c measures either a or b.Suppose c does not measure a.Therefore c, a are prime to one another. ［VII. 29］Suppose ab＝mc.Therefore c : a ＝ b : m. ［VII. 19］Hence ［VII. 20, 21］ b＝nc, where n is some integer.Therefore c measures b.Similarly, if c does not measure b, c measures a.Therefore c measures one or other of the two numbers a, b.Q.E.D. See also Bézout's identity Euclidean algorithm Fundamental theorem of arithmetic Irreducible element Prime element Prime number Footnotes Notes Citations References . - vol. 2 . . . External links Articles containing proofs Lemmas in number theory Theorems about prime numbers