Elements of Algebra

530 ADDITIONS. CHAP. III. CHAP. III. Of the Resolution, in Integer Numbers, of Equations of the first Degree, containing two unknown Quantities. [APPENDIX TO CHAP. I.] 42. When we have to resolve an equation of this form, $ax - by = c$, in which $a, b, c$, are given integer numbers, positive, or negative, and in which the two unknown quantities, $x$ and $y$, must also be integers, it is sufficient to know one solution, in order to deduce with ease all the other solutions that are possible. For, suppose we know that these values, $x = \alpha$, and $y = \beta$, satisfy the conditions of the equation proposed, $\alpha$ and $\beta$ being any integer numbers, we shall then have $a\alpha - b\beta = c$; and, consequently, $ax - by = a\alpha - b\beta$, or $a(x - \alpha) - b(y - \beta) = 0$; whence we find $\frac{x-\alpha}{y-\beta} = \frac{b}{a}$. Let us reduce the fraction $\frac{b}{a}$ to its least terms, and supposing, in consequence of this reduction, that it becomes $\frac{b'}{a'}$, where $b'$ and $a'$ will be prime to one another, it is evident that the equation, $\frac{x-\alpha}{y-\beta} = \frac{b'}{a'}$ could not subsist, on the supposition of $x - \alpha$, and $y - \beta$, being integers, unless we have $x - \alpha = mb'$, and $y - \beta = ma'$, $m$ being any integer number; so that we shall have, in general, $x = \alpha + mb'$, and $y = \beta + ma'$; $m$ being an indeterminate integer. Now, as we may take $m$ either positive, or negative, it is easy to perceive, that we may always determine the number $m$ in such a manner, that the value of $x$ may not be greater than $\frac{b'}{2}$, or that of $y$ not greater than $\frac{a'}{2}$, (abstracting from the signs of these quantities); whence it follows, that if the