G.252 numbers prime find — Heath

VII.3

Given three numbers not prime to one another, to find their greatest common measure.

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.

For let the greatest common measure, D, of the two numbers A, B be taken ; [G.251]

then D either measures, or does not measure, C.

A B C D E F

First, let it measure it.

But it measures A, B also ;
therefore D measures A, B, C ;
therefore D is a common measure of A, B, C.

I say that it is also the greatest.

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.

Let such a number measure them, and let it be E.

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. [G.251]
But the greatest common measure of A, B is D ;
therefore E measures D, the greater the less : which is impossible.
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.

Next, let D not measure C ;

I say first that C, D are not prime to one another.

For, since A, B, C are not prime to one another, some number will measure them.
Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B. [G.251]
But it measures C also ;
therefore some number will measure the numbers D, C ;
therefore D, C are not prime to one another.

Let then their greatest common measure E be taken. [G.251]

Then, since E measures D, and D measures A, B,
therefore E also measures A, B.
But it measures C also ;
therefore E measures A, B, C ;
therefore E is a common measure of A, B, C.

I say next that it is also the greatest.

For, if E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C.

Let such a number measure them, and let it be F.

Now, since F measures A, B, C, it also measures A, B ;
therefore it will also measure the greatest common measure of A, B. [G.251]
But the greatest common measure of A, B is D ;
therefore F measures D.
And it measures C also ;
therefore F measures D, C ;
therefore it will also measure the greatest common measure of D, C. [G.251]
But the greatest common measure of D, C is E ;
therefore F measures E, the greater the less : which is impossible.
Therefore no number which is greater than E will measure the numbers A, B, C ;
therefore E is the greatest common measure of A, B, C.
Q. E. D.

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Required for

G.289, G.290, G.292, G.295, G.298, G.299, G.301, G.303, G.304, G.305, G.314, G.315, G.316, G.317, G.322, G.323, G.324, G.325, G.326, G.327, G.328, G.329, G.330, G.331, G.332, G.333, G.334, G.335, G.337, G.338, G.339, G.342, G.343, G.364, G.370, G.381, G.383, G.384, G.385, G.389, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.403, G.404, G.405, G.406, G.407, G.408, G.409, G.410, G.411, G.412, G.413, G.414, G.415, G.416, G.417, G.418, G.419, G.420, G.421, G.423, G.425, G.427, G.429, G.431, G.433, G.434, G.435, G.436, G.437, G.438, G.439, G.440, G.441, G.442, G.443, G.444, G.445, G.446, G.447, G.448, G.449, G.450, G.451, G.452, G.453, G.457, G.458, G.459, G.460, G.461, G.462, G.463, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.478, G.479, G.480, G.481, G.482, G.483, G.484, G.485, G.486, G.487, G.488, G.489, G.490, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.498, G.499, G.500, G.584, G.589, G.597, G.600, G.601