Elements of Algebra

CHAP. I. OF ALGEBRA. 309 20) 31 (1 20 --- 11) 20 (1 11 --- 9) 11 (1 9 --- 2) 9 (4 8 --- 1) 2 (2 2 --- 0, it is evident that the quotients are found also in the successive values of the letters $p, q, r, s, \&c.$ and that they are connected with the first letter to the right, while the last always remains alone. We see, farther, that the number $7$ occurs only in the fifth and last equation, and is affected by the sign $+$, because the number of this equation is odd; for if that number had been even, we should have obtained $-7$. This will be made more evident by the following Table, in which we may observe the decomposition of the numbers $31$ and $20$, and then the determination of the values of the letters $p, q, r, \&c.$ $31 = 1 \times 20 + 11$ | $p = 1 \times q + r$ $20 = 1 \times 11 + 9$ | $q = 1 \times r + s$ $11 = 1 \times 9 + 2$ | $r = 1 \times s + t$ $9 = 4 \times 2 + 1$ | $s = 4 \times t + u$ $2 = 2 \times 1 + 0$ | $t = 2 \times u + 7$ 18. In the same manner, we may represent the example in Art. 14. $56 = 1 \times 39 + 17$ | $p = 1 \times q + r$ $39 = 2 \times 17 + 5$ | $q = 2 \times r + s$ $17 = 3 \times 5 + 2$ | $r = 3 \times s + t$ $5 = 2 \times 2 + 1$ | $s = 2 \times t + u$ $2 = 2 \times 1 + 0$ | $t = 2 \times u + 11$ 19. And, in the same manner, we may analyse all questions of this kind. For, let there be given the equation $bp = aq + n$, in which $a, b$, and $n$, are known numbers; then, we have only to proceed as we should do to find the greatest common divisor of the numbers $a$ and $b$, and we