Prop. 2. If the product ab be divisible by c, and if c be prime to b, it must divide a. Let
| ab | = d, then | b | = | d | . |
| c | c | c |
Now b/c is in its lowest terms; therefore, by the last proposition, d and a must have a common measure. Let the greatest common measure be k, and let a = kl, d = km. Then
| b | = | km | = | m | , and | m |
| c | kl | l | l |
is also in its lowest terms; but so is b/c; therefore we must have m = b, l = c, for otherwise a fraction in its lowest terms would be equal to another of lower terms. Therefore a = kc, or a is divisible by c. And from this it follows, that if a number be prime to two others, it is prime to their product. Let a be prime to b and c, then no measure of a can measure either b or c, and no such measure can measure the product bc; for any measure of bc which is prime to one must measure the other.
Prop. 3. If a be prime to b, it is prime to all the powers of b. Every measure[62] of a is prime to b, and therefore does not divide b. Hence, by the last, no measure of a divides b²; hence, a is prime to b², and so is every measure of it; therefore, no measure of a divides bb², consequently a is prime to b³, and so on.
Hence, if a be prime to b, a cannot divide without remainder any power of b. This is the reason why no fraction can be made into a decimal unless its denominator be measured by no prime[63] numbers except 2 and 5. For if
| a | = | c | , |
| b | 10ⁿ |
which last is the general form of a decimal fraction, let
| a | be in its lowest terms; then | 10ⁿa | , |
| b | b |