APPENDIX IX. ON SOME GENERAL PROPERTIES OF NUMBERS.

be one of the steps, and let a = kv + x, b = kw + y; so that

Now, if x = 0 but not y, this is absurd, for it gives

A similar absurdity follows if y be 0, but not x; and if both x and y be = 0, then a = kv, b = kw, or a and b have a common measure, k. Now k must be greater than 1, for v and w are less than c and d, which by hypothesis are less than a and b. Consequently a and b have a common measure k greater than 1, which by hypothesis they have not. If, then, a and b be integers not divisible by any integer greater than 1, the fraction a/b is really in its lowest terms. Also a and b are said to be prime to one another.

Prop. 2. If the product ab be divisible by c, and if c be prime to b, it must divide a. Let

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

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 measure² 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 prime³ numbers except 2 and 5. For if

which last is the general form of a decimal fraction, let