Elements of Algebra

532 ADDITIONS. CHAP. III. but one by $\frac{p}{q}$, we shall have, by the law of these fractions, (Art. 12) $a'p - b'q = \pm 1$; the upper sign being for the case, in which the rank of the fraction is even, and the under for that in which it is odd. These values of $p$ and $q$ being thus known, we shall first have $x = \pm pc'$, and $y = \pm qc'$, and then taking these values for $\alpha$ and $\beta$, we shall have, in general, (Art. 42), $x = \pm pc' + mb'$, $y = \pm qc' + ma'$, expressions which necessarily include all the solutions of the given equation that are possible in integer numbers. That we may leave no obstacle to the practice of this method, we shall observe, that although the numbers $a$ and $b$ may be positive, or negative, we may notwithstanding take them always positive, provided we give contrary signs to $x$, when $a$ is negative, and to $y$, when $b$ is negative. 44. Example. To give an example of the preceding method, we shall take that of Art. 14, Chap. I. of the preceding Treatise, where it is required to resolve the equation, $39p = 56q + 11$. Changing $p$ into $x$, and $q$ into $y$, we shall have $39x - 56y = 11$. So that we shall make $a = 39$, $b = 56$, and $c = 11$; and as 56 and 39 are already prime to each other, we shall have $a' = 39$, $b' = 56$, $c' = 11$. We must therefore reduce the fraction $\frac{b'}{a'} = \frac{56}{39}$ to a continued fraction; and, for this purpose, as we have already done (Art. 20), we shall make the following calculation: 39)56(1 39 -- 17)39(2 34 -- 5)17(3 15 -- 2)5(2 4 -- 1)2(2 2 -- 0.