If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. Any prime number is relatively prime to any number which it does not measure. If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. Any composite number is measured by some prime number.
Any number is either prime or is measured by some prime number.
Given as many numbers as we please, to find the least of those which have the same ratio with them. Proposition 34 To find the least number which two given numbers measure. If two numbers measure any number, then the least number measured by them also measures the same. Proposition 36 To find the least number which three given numbers measure. If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.