G.252 numbers prime find

VII.3

Given three numbers not prime to one another, to find their greatest common measure.
numbers_prime_find_vii_3(A, B, C) → E
given A, B, C be the three given numbers not prime to one another
construct E is the greatest common measure of A, B, C
A B C D E F
given A, B, C be the three given numbers not prime to one another
construct numbers_prime_find: Let the greatest common measure, D, of the two numbers A, B be taken
assert Then D either measures, or does not measure, C
assert First, let it measure it
assert It measures A, B also
assert D measures A, B, C
assert D is a common measure of A, B, C
assert If D is not the greatest common measure of A, B, C, some number which > D will measure the numbers A, B, C
assert Let such a number measure them, and let it be E
assert then E measures A, B, C, it will also measure A, B
∵ numbers_prime_findassert it will also measure the greatest common measure of A, B
assert The greatest common measure of A, B is D
assert E measures D, the greater the less ⇒⇐
assert no number which > D will measure the numbers A, B, C
assert D is the greatest common measure of A, B, C
assert Next, let D not measure C
assert I say first that C, D are not prime to one another
assert A, B, C are not prime to one another, some number will measure them
∵ numbers_prime_findassert Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B
assert It measures C also
assert some number will measure the numbers D, C
assert D, C are not prime to one another
construct numbers_prime_find: Let then their greatest common measure E be taken.
assert E measures D, and D measures A, B,
assert E also measures A, B
assert It measures C also
assert E measures A, B, C
assert E is a common measure of A, B, C
assert I say next that it is also the greatest
assert If E is not the greatest common measure of A, B, C, some number which > E will measure the numbers A, B, C
assert Let such a number measure them, and let it be F
assert F measures A, B, C, it also measures A, B
∵ numbers_prime_findassert it will also measure the greatest common measure of A, B
assert The greatest common measure of A, B is D
assert F measures D
assert It measures C also
assert F measures D, C
∵ numbers_prime_findassert it will also measure the greatest common measure of D, C
assert The greatest common measure of D, C is E
assert F measures E, the greater the less ⇒⇐
assert no number which > E will measure the numbers A, B, C
∴ E is the greatest common measure of A, B, C

Dependency Graph

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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