PROPOSITION 34.
Given two numbers, to find the least number which they measure.
Let
A,
B be the two given numbers; thus it is required to find the least number which they measure.
Now
A,
B are either prime to one another or not.
First, let
A,
B be prime to one another, and let
A by multiplying
B make
C; therefore also
B by multiplying
A has made
C. [
VII. 16]
Therefore
A,
B measure
C
I say next that it is also the least number they measure.
For, if not,
A,
B will measure some number which is less than
C.
Let them measure
D.
Then, as many times as
A measures
D, so many units let there be in
E, and, as many times as
B measures
D, so many units let there be in
F; therefore
A by multiplying
E has made
D, and
B by multiplying
F has made
D; [
VII. Def. 15] therefore the product of
A,
E is equal to the product of
B,
F.
Therefore, as
A is to
B, so is
F
E. [
VII. 19]
But
A,
B are prime, primes are also least, [
VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [
VII. 20] therefore
B measures
E, as consequent consequent.
And, since
A by multiplying
B,
E has made
C,
D, therefore, as
B is to
E, so is
C to
D. [
VII. 17]
But
B measures
E; therefore
C also measures
D, the greater the less: which is impossible.
