PROPOSITION 23.

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

Let A, B be two numbers prime to one another, and let any number C measure A;

I say that C, B are also prime to one another.

A B C D
For, if C, B are not prime to one another, some number will measure C, B.

Let a number measure them, and let it be D.

Since D measures C, and C measures A,
therefore D also measures A.
But it also measures B ;
therefore D measures A, B which are prime to one another : which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers C, B.
Therefore C, B are prime to one another.
Q. E. D.

Dependency Graph

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Required for

VII.24, VII.25, VII.26, VII.27, VIII.2, VIII.3, VIII.6, VIII.7, VIII.8, VIII.9, VIII.14, VIII.15, VIII.16, VIII.17, VIII.21, VIII.23, VIII.24, VIII.25, VIII.26, VIII.27, IX.1, IX.2, IX.3, IX.4, IX.5, IX.6, IX.8, IX.9, IX.10, IX.12, IX.13, IX.15, IX.32, IX.36, X.9, X.10, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.39, X.40, X.41, X.45, X.46, X.48, X.49, X.50, X.51, X.52, X.53, X.57, X.58, X.59, X.60, X.63, X.64, X.65, X.68, X.69, X.70, X.71, X.72, X.76, X.77, X.78, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.94, X.95, X.96, X.100, X.101, X.102, X.105, X.106, X.107, X.108, X.109, X.110, X.111, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition VII.23.