PROPOSITION 15.
If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will be prime to the remaining number.
Let
A,
B,
C, three numbers in continued proportion, be the least of those which have the same ratio with them; I say that any two of the numbers
A,
B,
C whatever added together are prime to the remaining number, namely
A,
B to
C;
B,
C to
A; and further
A,
C to
B.
For let two numbers
DE,
EF, the least of those which have the same ratio with
A,
B,
C, be taken. [
VIII. 2]
It is then manifest that
DE by multiplying itself has made
A, and by multiplying
EF has made
B, and, further,
EF by multiplying itself has made
C. [
VIII. 2]
Now, since
DE,
EF are least, they are prime to one another. [
VII. 22]
But, if two numbers be prime to one another, their sum is also prime to each; [
VII. 28] therefore
DF is also prime to each of the numbers
DE,
EF.
But further
DE is also prime to
EF; therefore
DF,
DE are prime to
EF.
But, if two numbers be prime to any number, their product is also prime to the other; [
VII. 24] so that the product of
FD,
DE is prime to
EF; hence the product of
FD,
DE is also prime to the square on
EF. [
VII. 25]
But the product of
FD,
DE is the square on
DE together with the product of
DE,
EF; [
II. 3] therefore the square on
DE together with the product of
DE,
EF is prime to the square on
EF.
And the square on
DE is
A, the product of
DE,
EF is
B, and the square on
EF is
C; therefore
A,
B added together are prime to
C.