If two numbers be prime to one another, the sum will also be prime to each of them ; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.
For let two numbers / A B /, / B C / prime to one another be added ;
I say that the sum / A C / is also prime to each of the numbers / A B /, / B C /.
For, if / A C /, / A B / are not prime to one another, some number will measure/ A C /, / A B /.
Let a number measure them, and let it be D.
Since then D measures / A C /, / A B /,
therefore it will also measure the remainder / B C /.
But it also measures / A B / ;
therefore D measures / A B /, / B C / which are prime to one another : which is impossible. [G.249]
Therefore no number will measure the numbers / A C /, / A B / ; therefore / A C /, / A B / are prime to one another.
For the same reason
/ A C /, / B C / are also prime to one another.
Therefore / A C / is prime to each of the numbers / A B /, / B C /.
Again, let / A C /, / A B / be prime to one another ;
I say that / A B /, / B C / are also prime to one another.
For, if / A B /, / B C / are not prime to one another, some number will measure / A B /, / B C /.
Let a number measure them, and let it be D.
Now, since D measures each of the numbers / A B /, / B C /,
it will also measure the whole / A C /.
But it also measures / A B / ;
therefore D measures / A C /, / A B / which are prime to one another : which is impossible. [G.249]
Therefore no number will measure the numbers / A B /, / B C /.
Therefore / A B /, / B C / are prime to one another.
