Book 7
31
Thus, if the investigation be continued in this way, some prime number will be found which will measure the number before it, which will also measure A.
31
For, if it is not found, an infinite series of numbers will measure the number A, each of which is less than the other: which is impossible in numbers.
31
Therefore some prime number will be found which will measure the one before it, which will also measure A.
31
Therefore any composite number is measured by some prime number. i.e. the implied problem of finding a prime number which measures A. In the Greek the sentence stops here, but it is necessary to add the words the number before it, which will also measure A, which are found a few lines further down. It is possible that the words may have fallen out of P here by a simple mistake due to ὁμοιοτέλευτον (Heiberg).
PROPOSITION 32.
32
Any number either is prime or is measured by some prime number.
32
Let A be a number; I say that A either is prime or is measured by some prime number.
32
If now A is prime, that which was enjoined will have been done.
32
But if it is composite, some prime number will measure it. [VII. 31]
32
Therefore any number either is prime or is measured by some prime number. Q. E. D.
PROPOSITION 33.
33
Given as many numbers as we please, to find the least of those which have the same ratio with them.
33
Let A, B, C be the given numbers, as many as we please; thus it is required to find the least of those which have the same ratio with A, B, C.
33
A, B, C are either prime to one another or not.
33
Now, if A, B, C are prime to one another, they are the least of those which have the same ratio with them. [VII. 21]
33
But, if not, let D the greatest common measure of A, B, C be taken, [VII. 3] and, as many times as D measures the numbers A, B, C respectively, so many units let there be in the numbers E, F, G respectively.
33
Therefore the numbers E, F, G measure the numbers A, B, C respectively according to the units in D. [VII. 16]
