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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.

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.

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