Let a number measure them, and let it be D. Now A is odd; therefore D is also odd. And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [IX. 30] But B is half of C; therefore D measures B. But it also measures A; therefore D measures A, B which are prime to one another: which is impossible. Therefore A cannot but be prime to C. Therefore A, C are prime to one another. Q. E. D.
Let a number measure them, and let it be D. Now A is odd; therefore D is also odd. And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [IX. 30] But B is half of C; therefore D measures B. But it also measures A; therefore D measures A, B which are prime to one another: which is impossible. Therefore A cannot but be prime to C. Therefore A, C are prime to one another. Q. E. D.
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