_ab_ _b_ _d_ ---- = _d_, then --- = ---. _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_ _m_ --- = ---- = ---, and --- _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 _b_²; hence, _a_ is prime to _b_², and so is every measure of it; therefore, no measure of _a_ divides _bb_², consequently _a_ is prime to _b_³, 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_ 10ⁿ_a_ --- be in its lowest terms; then ------ _b_ _b_ is an integer, whence (Prop. 2) _b_ must divide 10ⁿ, and so must all the divisors of _b_. If, then, among the divisors of _b_ there be any prime numbers except 2 and 5, we have a prime number (which is of course a number prime to 10) not dividing 10, but dividing one of its powers, which is absurd. [62] For that which measures a measure is itself a measure; so that if a measure of _a_ could have a measure in common with _b_, _a_ itself would have a common measure with _b_. [63] A prime number is one which is prime to all numbers except its own multiples, or has no divisors except 1 and itself. PROP. 4. If _b_ be prime to _a_, all the multiples of _b_, as _b_, 2_b_, ... up to (_a_-1)_b_ must leave different remainders when divided by _a_. For if, _m_ being greater than _n_, and both less than _a_, we have _mb_ and _nb_ giving the same remainder, it follows that _mb_-_nb_, or (_m_-_n_)_b_, is divisible by _a_; whence (Prop. 2), a divides _m_-_n_, a number less than itself, which is absurd. * * * * *