PROPOSITION 36.
Given three numbers, to find the least number which they measure.
Let
A,
B,
C be the three given numbers; thus it is required to find the least number which they measure.
Let
D, the least number measured by the two numbers
A,
B, be taken. [
VII. 34]
Then
C either measures, or does not measure,
D.
First, let it measure it.
But
A,
B also measure
D; therefore
A,
B,
C measure
D.
I say next that it is also the least that they measure.
For, if not,
A,
B,
C will measure some number which is less than
D.
Let them measure
E.
Since
A,
B,
C measure
E, therefore also
A,
B measure
E.
Therefore the least number measured by
A,
B will also measure
E. [
VII. 35]
But
D is the least number measured by
A,
B; therefore
D will measure
E, the greater the less: which is impossible.
Therefore
A,
B,
C will not measure any number which is less than
D;
therefore D is the least that A, B, C measure.
Again, let
C not measure
D, and let
E, the least number measured by
C,
D, be taken. [
VII. 34]
Since
A,
B measure
D, and
D measures
E, therefore also
A,
B measure
E.
But
C also measures
E; therefore also
A,
B,
C measure
E.
I say next that it is also the least that they measure.
For, if not,
A,
B,
C will measure some number which is less than
E.
Let them measure
F.
Since
A,
B,
C measure
F, therefore also
A,
B measure
F; therefore the least number measured by
A,
B will also measure
F. [
VII. 35]
But
D is the least number measured by
A,
B; therefore
D measures
F.
But
C also measures
F; therefore
D,
C measure
F, so that the least number measured by
D,
C will also measure
F.
But
E is the least number measured by
C,
D; therefore
E measures
F, the greater the less: which is impossible.
Therefore
A,
B,
C will not measure any number which is less than
E.
Therefore
E is the least that is measured by
A,
B,
C. Q. E. D.