Book 7
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But it also measures the whole BA; therefore it will also measure the remainder AE.
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But AE measures DF; therefore G will also measure DF.
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But it also measures the whole DC; therefore it will also measure the remainder CF, that is, the greater will measure the less: which is impossible.
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Therefore no number which is greater than CF will measure the numbers AB, CD; therefore CF is the greatest common measure of AB, CD.
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Porism. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.
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Q. E. D.
PROPOSITION 3.
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Given three numbers not prime to one another, to find their greatest common measure.
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Let A, B, C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A, B, C.
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For let the greatest common measure, D, of the two numbers A, B be taken; [VII. 2] then D either measures, or does not measure, C.
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First, let it measure it.
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But it measures A, B also; therefore D measures A, B, C; therefore D is a common measure of A, B, C.
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I say that it is also the greatest.
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For, if D is not the greatest common measure of A, B, C, some number which is greater than D will measure the numbers A, B, C.
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Let such a number measure them, and let it be E.
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Since then E measures A, B, C, it will also measure A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]
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But the greatest common measure of A, B is D; therefore E measures D, the greater the less: which is impossible.
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Therefore no number which is greater than D will measure the numbers A, B, C; therefore D is the greatest common measure of A, B, C.
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Next, let D not measure C; I say first that C, D are not prime to one another.
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For, since A, B, C are not prime to one another, some number will measure them.
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Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B. [VII. 2, Por.]
