b than b3, of c than itself, will divide the quantities; and that d will not divide them; therefore, the divisor sought is 4 a3 b3 c. Hence, RULE. Annex to the greatest common divisor of the coefficients those letters which are common to all the quantities, giving to each letter the least exponent it has in any of the quantities. 2. Find the greatest common divisor of 63 a3 b1 c1 d®, 27 a1 b3 c5, and 45 a2 bo c3 d. Ans. 9 a2 b c3. 3. Find the greatest common divisor of 75 x1 y5 28 and 125 a b x3 y1 z3. 4. Find the greatest common divisor of 99 a b2 c1 d3 x® y5 and 22 a2 b1 c3 do x5. Ans. 11 a b2 c3 d3 x5. 5. Find the greatest common divisor of 17 x1 y2, 19 x2 y3, and 212 bx y1 z5. CASE II. 76. To find the greatest common divisor of polynomials. 1. Find the greatest common divisor of a2 — y2 and x2-2xy + y2. Arrange the terms of both quantities in the order of the powers of some letter, and then proceed according to the general rule in Art. 74. NOTE 1.-If the leading term of the dividend is not divisible by the leading term of the divisor, it can be made so by introducing in the dividend a factor which contains no factor of the divisor; or either quantity may be simplified by rejecting any factor which contains no factor of the other. (Art. 74, Note 3.) a, a, NOTE 2. Since any quantity which will divide a will divide and vice versa, and any quantity divisible by a is divisible by and vice versa, therefore all the signs of either divisor or dividend, or of both, may be changed from + to or to +, without changing the common divisor. NOTE 3. When one of the quantities is a monomial, and the other a polynomial, either of the given rules can be applied, although generally the greatest common divisor will be at once apparent. 2. Find the greatest common divisor of a x7. 8a5x8 2 acx3+4a2 c x2 6 a3 c x 20 a1 c. 8 a5 x3) 2 c x2 - 2 a cx2+4a2c x2- 6 a3 c x 20 a*c ах 3. Find the greatest common divisor of a — x1 and a3 + a2 xa x2 — x3. Ans. a2 - x2. 4. Find the greatest common divisor of a1 a5 — a3 x2. x1 and Ans. a2x2. a2x — a3 5. Find the greatest common divisor of 2 a x2. and 2x2+3 a x + a2. 2x+a 6. Find the greatest common divisor of 6 a x and 6 a xa x2 12 a x. Ans. 3 ax 7. Find the greatest common divisor of x* x2+y3. 8. Find the greatest common divisor of 3 x3 and 2x3 16 x 6. y1 and 9. Find the greatest common divisor of x3 x2- y2. 10. Find the greatest common divisor of 10 — 20 x2 y +30 y3 and x3 + 2 x2 y + 2 x y2 + y3. Ans. x+y. 11. Find the greatest common divisor of a + a3 + a2 +a-4 and a* + 2 a3 + 3 a2 + 4 a 10. 12. Find the greatest common divisor of 7 a x2 + 21 a x3 13. Find the greatest common divisor of 27 a3 ya . and 3 y 2 ay + 3 a2 y — 2 a3 y1. Ans. 3y-2ay. 10 14. Find the greatest common divisor of a3 + a and a1 16. NOTE 5. The greatest common divisor of polynomials can also be found by factoring the polynomials, and finding the product of the factors common to the polynomials, taking each factor the least number of times it occurs in any of the quantities. (Art. 74, Note 4.) 15. Find the greatest common divisor of 3 a x2 — 4 a x + Заху 4 a and a3 x y x + a3 y y. 3 a x2-4a x+3axy-4ay = a(x + y) (3x-4) a3 x x + a3 y − y = (x + y) (a − 1) (a2 + a + 1) Ans. xy. SECTION XI.. LEAST COMMON MULTIPLE. 77. A MULTIPLE of any quantity is a quantity that can be divided by it without remainder. 78. A COMMON MULTIPLE of two or more quantities is any quantity that can be divided by each of them withDut remainder. 79. The LEAST COMMON MULTIPLE of two or more quantities is the least quantity that can be divided by each of them without remainder. 80. It is evident that a multiple of any quantity must contain the factors of that quantity; and, vice versa, any quantity that contains the factors of another quantity is a multiple of it: and a common multiple of two or more quantities must contain the factors of these quantities; and the least common multiple of two or more quantities must contain only the factors of these quantities. CASE I. To find the least common multiple of monomials. 1. Find the least common multiple of 6 a2 b2 c, 8 a3 b3 c2 d, and 12 a b c x. The least common multiple of the coefficients, found by inspection or the rule in Arithmetic, is 24; it is evident that no quantity which contains a power of a less than at, of b less than 65, of c less than c2, and which does not contain d and x, can be divided by each of these quantities; therefore the multiple sought is 24 aa b3 c2 d x. Hence, in the case of monomials, RULE. Annex to the least common multiple of the coefficients all the letters which appear in the several quantities, giving to each letter the greatest exponent it has in any of the quantities. 2. Find the least common multiple of 3 at b2 co, 6 a2 b1 c d2, and 10 abc x5. Ans. 30 a ba cô d2 x5. 3. Find the least common multiple of 16 a b x, 80 a b1 x2, and 35 a b x1. Ans. 560 a ba x1. 4. Find the least common multiple of 9 a3 b5, 15 a* b x, and 18 a x y2. Ans. 90 at b5 6 y2. 5. Find the least common multiple of 18 a3 b c1 x, 24 a b c x2y, and 30 a2 b2 x z. 6. Find the least common multiple of 100 x y z, 45 a b c, and 25 m n. 7. Find the least common multiple of 10 a2 by2, 13 aa b2 c, and 17 a2 b3 c2. 8. Find the least common multiple of 14 a3 b2 ca, 20 a b c1, 25 a b c3, and 28 a b c d. CASE II. 81. To find the least common multiple of any two quantities. Since the greatest common divisor of two quantities contains all the factors common to these quantities (Art. 74, Note 4); and since the least common multiple of two quantities must contain only the factors of these quantities (Art. 80); if the product of two quantities is divided by their greatest common divisor, the quotient will be their least common multiple. Hence, to find the least common multiple of any two quantities, |