G.291 numbers measure number — Heath

VII.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 / C D /, and let E be the least that they measure;

I say that E also measures / C D /.

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

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Required for

G.292, G.295, G.299, G.301, G.384, G.385, G.389, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, 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.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