PROP. 1. If a fraction be reduced to its lowest terms, _so called_,[61] that is, if neither, numerator nor denominator be divisible by any integer greater than unity, then no fraction of a smaller numerator and denominator can have the same value. [61] This theorem shews that what is _called_ reducing a fraction to its lowest terms (namely, dividing numerator and denominator by their greatest common measure), is correctly so called. Let _a_/_b_ be a fraction in which _a_ and _b_ have no common measure greater than unity: and, if possible, let _c_/_d_ be a fraction of the same value, _c_ being less than _a_, and _d_ less than _b_. Now, since _a_ _c_ _a_ _b_ --- = ---, we have --- = ---; _b_ _d_ _c_ _d_ let _m_ be the integer quotient of these last fractions (which must exist, since _a_ > _c_, _b_ > _d_), and let _e_ and _f_ be the remainders. Then _a_ _mc_ + _e_ _c_ _mc_ --- or ---------- = --- = ---- _b_ _md_ + _f_ _d_ _md_ Hence, _e_ _mc_ --- and ---- must be equal, for if not, _f_ _md_ _mc_ + _e_ _mc_ _e_ ---------- would lie between ---- and ---, _md_ + _f_ _md_ _f_ instead of being equal to the former. Hence, _a_ _e_ --- = ---; _b_ _f_ so that if a fraction whose numerator and denominator have no common measure greater than unity, be equal to a fraction of lower numerator and denominator, it is equal to another in which the numerator and denominator are still lower. If we proceed with _a_ _e_ --- = --- in a similar manner, we find _b_ _f_ _a_ _g_ --- = --- where _g_ < _e_, _h_ < _f_, _b_ _h_ and so on. Now, if there be any process which perpetually diminishes the terms of a fraction by one or more units at every step, it must at last bring either the numerator or denominator, or both, to 0. Let _a_ _v_ --- = --- _b_ _w_ be one of the steps, and let _a_ = _kv_ + _x_, _b_ = _kw_ + _y_; so that _kv_ + _x_ _v_ ---------- = ---. _kw_ + _y_ _w_ Now, if _x_ = 0 but not _y_, this is absurd, for it gives _kv_ _kv_ ---------- = ----. _kw_ + _y_ _kw_ 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