1. The difference of the same powers of two quantities is divisible by the difference of the quantities. Let a and b represent two quantities and a > b, and by actual division we find II. The difference of the same even powers of two quantities is divisible by the sum of the quantities. It follows from the two preceding statements that The difference of the same even powers of two quantities is divisible by either the sum or the difference of the quan tities. III. The sum of the same odd powers of two quantities is divisible by the sum of the quantities. (x3 — y3) ÷ (x − y) = x* + x3 y + x2y2 + xy3 + y^ By I. of this article, the difference of the same powers of two quantities is divisible by the difference of the quantities; therefore y must be a factor of y; and dividing x — y by x — gives the other factor x + x3 y + x2 y2+ x y3+y*. x 2. Find two factors of co — do. OPERATION. c5 x-y (cd) ÷ (c+d) = c1 — c1 d + c3 d2 — c2 d3 + c d1 — d3 By II. the difference of the same even powers of two quantities is divisible by the sum of the quantities; therefore c+d must be a factor of cd; and dividing c d by cd gives the other factor c5 · c2 d + c3 d2 — c2 d3 + c d1 — d3. 3. Find the factors of m3 + n3. OPERATION. (m3 + n3) ÷ (m + n) = m2 — m3 n + m2 n2 — m n3 + n2 By III. the sum of the same odd powers of two quantities is divisible by the sum of the quantities; therefore m+n must be a factor of mn; and dividing m3 + no by m + n gives the other factor mt m3 n + m2 n3· m n3 + n*. NOTE. -In Example 2, the factors of c d there obtained are not the only factors; for by I. c. d' is divisible by cd; and dividing c by c- d gives another factor, - do or by Art. 70, c3 + c1d + c3 d2 + c2 d3 + c d'+d3 ; c6d = · (c3 + d3) (c3 — d3). are not prime quantities; for the first can be divided by c— - d, and the quotient thus arising can be divided by c2± cd+d2; the second can be divided by c+d, and the quotient thus arising will be the same as after the division of the first quantity by c d, and can be divided by c2±cd+d2; the third can be divided by c+d, and the fourth by cd. Performing these divisions, by each method we shall find the prime factors of c -de to be c+d, c d, c2 + c d + d2, and c2 -cd+d2. In finding the prime factors, it is better to apply first the principle of Art. 70 as far as possible. = (x + y) (x2 - x3 y + x2 y2 — x y3 + y1). = (x − y) (∞x2 + x23 y + x2 y2 + x y3 + y1). - Ans. (x+y) (xy) (x2 — x3y+x2 y2 — x y3 + y1) (x2 + x3 y + x2 y2+ x y3 +y*). 7. Find the prime factors of a® — 1. Ans. (a+1)(a− 1) (a2+a+1) (a2 —a+1). 8. Find the prime factors of a 2 a2x2+x1. Ans. (a + x) (a + x) (a — x) (a — x). 9. Find the prime factors of x2 + 2 x3 y3 + yo. Ans. (x+y) (x+y) (x2−xy+y2) (x2 —xy+y2). SECTION X. GREATEST COMMON DIVISOR.* 72. A COMMON DIVISOR of two or more quantities is any quantity that will divide each of them without remainder. 73. THE GREATEST COMMON DIVISOR of two or more quantities is the greatest quantity that will divide each of them without remainder. 74. To deduce a rule for finding the greatest common divisor of two or more quantities, we demonstrate the two following theorems: THEOREM I. A common divisor of two quantities is also a common divisor of the sum or the difference of any multiples of each. Let A and B be two quantities, and let d be their common divisor; d is also a common divisor of m A±n B. That is, d is contained in m A + n B, m pn q times, and in m A n B, mp―n q times; i. e. d is a common divisor of the sum or the difference of any multiples of A and B. THEOREM II. The greatest common divisor of two quantities is also the greatest common divisor of the less and the remainder after dividing the greater by the less. Let A and B be two quantities, and A > B; and let the process of dividing be as appears in B) A (q the margin. Then, as the dividend is equal to the product of the divisor by the quotient plus 9 B the remainder, A = (1) *See Preface. And, as the remainder is equal to the dividend minus the product of the divisor by the quotient, r A 1 B. (2) Therefore, according to the preceding theorem, from (1) any divisor of r and B must be a divisor of A ; and from (2) any divisor of A and B, a divisor of r; i. e. the divisors of A and B and B and r are identical, and therefore the greatest common divisor of A and B must also be the greatest common divisor of B and r. In the same way the greatest common divisor of B and r is the greatest common divisor of r and the remainder after dividing B by r. Hence, to find the greatest common divisor of any two quantities, RULE. Divide the greater by the less, and the less by the remainder, and so continue till the remainder is zero; the last divisor is the divisor sought. NOTE 1. The division by each divisor should be continued until the remainder will contain it no longer. NOTE 2. If the greatest common divisor of more than two quantities is required, find the greatest common divisor of two of them, then of this divisor and a third, and so on; the last divisor will be the divisor sought. NOTE 3. -The common divisor of x y and xz is x; x is also the common divisor of x and xz, or of a x y and Xz; i. e. the common divisor of two quantities is not changed by rejecting or introducing into either any factor which contains no factor of the other. NOTE 4. It is evident that the greatest common divisor of two quantities contains all the factors common to the quantities. 75. To find the greatest common divisor of monomials. 1. Find the greatest common divisor of 8 a2 b c d, 16 a3 3 c2, and 28 a1 b* c. The greatest common divisor of the coefficients found by the general rule is 4; it is evident that no higher power of a than a2, of |