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 without 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 b5 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 a*, of b less than b3, 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 a1 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 quan tities. 2. Find the least common multiple of 3 a1 b2 co, 6 a2 ba c d2, and 10 a b c x5. Ans. 30 a7 b4 c6 d2 x5. 3. Find the least common multiple of 16 a b x, 80 a ba x2, and 35 a b x1. Ans. 560 a7 ba x1. 4. Find the least common multiple of 9 a3 65, 15 aa b xo, and 18 ax y2. Ans. 90 at b5 x y2. 5. Find the least common multiple of 18 a3 b c1 x, 24 a b3 с y, 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 a2 b ca, 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, RULE. Divide one of the quantities by their greatest common divisor, and multiply this quotient by the other quantity, and the product will be their least common multiple. NOTE 1. If the least common multiple of more than two quantities is required, find the least common multiple of two of them, then of this common multiple and a third, and so on; the last common multiple will be the multiple sought. NOTE 2. - In case the least common multiple of several monomials and polynomials is required, it may be better to find the least common multiple of the monomials by the Rule in Case I., and of the polynomials by the Rule in Case II., and then the least common multiple of these two multiples by the latter Rule. 1. Find the least common multiple of x2 x2-2xy + y2. OPERATION. xy) x2 - 2 x y + y2 х y (x2 - y2) (x − y), Ans. - y2 and y, with which we divide one of the quantities; and multiplying the other quantity by this quotient, we have the least common multiple (x2 — y3) (x − y). 2. Find the least common multiple of 2 a2x2, a1- x1, and a5 a3 x2. 4 x23 y, The least common multiple of the monomials is 4 a2x2y; and the least common multiple of the polynomials is a3 (a1 — x1). The greatest common divisor of these two multiples is a2; and dividing one of these multiples by a2, and multiplying the quotient by the other, we have 4 a3 x2 y (a* — x1) as the least common multiple. 3. Find the least common multiple of 3 a2 68, 6 a2 by, a88, and a2-4a4. Ans. 6 a2 by (a3 — 8) (a — 2). 5. Find the least common multiple of a-x and 6. Find the least common multiple of x and (x-1)2. a1 Ans. x6 ·y1 and x3 + y3. 7. Find the least common multiple of x1 8. Find the least common multiple of a3 + a NOTE 3.- The least common multiple of any quantities can also be found by factoring the quantities, and finding the product of all the factors of the quantities, taking each factor the greatest number of times it occurs in any of the quantities. (Art 80.) 9. Find the least common multiple of x2 xy, and (x + y)2. -2xy + y2, x1 — y1 = (x2 + y2) (x + y) (x − y) (x + y)2= (x + y) (x + y) Hence L. C. M = (x − y) (x − y) (x2 + y2) (x+y) (x+y) 10 3 axy Find the least common multiple of 3 a x2 — 4 a x + 4 ay and a3 x x + a3 y — y. (See 15th Example, Art. 76.) Ans. a(x+y) (3x-4) (a2 + a + 1) (a-1)=3a+x24a+x+3axy-4a1y - 3ax2+4ax-3axy+4ay. SECTION XII. FRACTIONS. 82. WHEN division is expressed by writing the dividend over the divisor with a line between, the expression is called a FRACTION. As a fraction, the dividend is called the numerator, and the divisor the denominator. Hence, the value of a fraction is the quotient arising from dividing the numerator by the denominator. xy Thus, is a fraction whose numerator is xy and denominator y, y and whose value is x. 83. The principles upon which the operations in fractions are carried on are included in the following THEOREM. Any multiplication or division of the numerator causes a like change in the value of the fraction, and any multiplication or division of the denominator causes an opposite change in the value of the fraction. which is y times the value of the given fraction. Dividing the numerator by y, which is y x y X, of the value of the given fraction. |