PROPOSITION 35.

If two numbers measure any number, the least number measured by them will also measure the same.

For let the two numbers A, B measure any number CD, and let E be the least that they measure;

I say that E also measures CD.

A B C F D E
For, if E does not measure CD, let E, measuring DF, leave CF less than itself.
Now, since A, B measure E, and E measures DF,
therefore A, B will also measure DF.
But they also measure the whole CD;
therefore they will also measure the remainder CF which is less than E: which is impossible.
Therefore E cannot fail to measure CD; therefore it measures it.
Q. E. D.

Dependency Graph

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Required for

VII.36, VII.39, VIII.4, VIII.5, X.12, X.13, X.17, X.18, X.22, X.23, X.24, X.25, X.26, X.27, X.28, X.31, X.32, X.33, X.34, X.35, X.36, X.37, X.38, X.39, X.40, X.41, X.42, X.43, X.44, X.45, X.46, X.47, X.48, X.49, X.50, X.52, X.53, X.54, X.55, X.56, X.57, X.58, X.59, X.60, X.61, X.62, X.63, X.64, X.65, X.66, X.67, X.68, X.69, X.70, X.71, X.72, X.73, X.75, X.76, X.77, X.78, X.79, X.80, X.81, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.92, X.93, X.94, X.95, X.96, X.97, X.98, X.99, X.100, X.101, X.102, X.103, X.104, X.105, X.106, X.107, X.108, X.109, X.110, X.111, X.112, X.113, X.114, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition VII.35.