120. Any expression which another will exactly divide is a multiple of it. 121. If two expressions have no common factor their least common multiple will be their product, since it will contain each of them. For brevity, let L. C. M. be used for Least Common Multiple. 122. When the Expressions can be Readily Factored. If the expressions can be readily factored their L. C. M. will be an expression which is the product of every factor of each expression taken the greatest number of times it occurs in any one of the given expressions. 1. Find the L. C. M. of 18 ar, 90ay, 12 axy. 2. Find the L. C. M. of xy- xy3, 3 x(x − y)2, 4 y(x — y)3. The rule can be more simply stated: The L. C. M. of two or more expressions which can be readily factored is the product of each factor taken the greatest number of times it occurs in any one of the expressions. 123. When the Quantities are not Readily Factored.-The L. C. M. of two or more quantities can be found by finding their G. C. D. Suppose that D is the G. C. D. of A and B; then Since D is G. C. D. of A and B, a and b have no common factor, The L. C. M. of two expressions can be found by dividing their product by their G. C. D.; or by a method which is usually more simple; by dividing one of the quantities by their G. C. D. and multiplying this quotient by the other. EXAMPLE. Find the L. C. M. of 20x + x2 — 1, and 25x5 x3 — x — 1. ... 5-1 is G. C. D. required. 20x15x-1=4x2 + 1. Hence, L. C. M. required is (4x2 + 1)(25 x1 + 5 æ3 — x − 1). 124. To find the L. C. M. of three quantities, A, B, and C; find the L. C. M., say M, of A and B; then the L. C. M. of M and C is the L. C. M. required, say M'. For M will contain each factor of A and B the greatest number of times it occurs in either of them; and M' will contain each factor of M and the greatest number of times it occurs in either M or C. Therefore, M' will contain each factor of A, B, and C the greatest number of times it occurs in A or B or C; and that is what is required to be proved. 125. For an understanding of the remaining chapters in this book. complete theories of the greatest common divisor and the least common multiple are not necessary. Such discussions will be found in works on the theory of equations. However, the solution of the examples illustrating the principles discussed in the preceding sections is a valuable exercise in the fundamental operation of algebra. 12. a(x-b) (x — c), b(c—x) (x − a), c(a− x) (b−x). 13. 1+y+ y2, 1−y+y2, 1+ y2+y1. 14. (1−x), (1 − x)2, (1—x)3. 15. (a+c)-2, (a+b)2 - c2, (b+c)2 — a2. 16. 17. 25+2+3+8x2+16x+8 and -4x3+x2-4. 18. 22+2x-1, 33-4x+1, and 2.3-3x+1. CHAPTER X FRACTIONS For 126. If a and b are two integral numbers such that the group of things represented by b can not be counted out of the group of things represented by a, then the symbol is a fraction. I. example, a is called the numerator and b the denominator of the fraction a b 5 The numerator and the denominator of the fraction are called the terms of the fraction. The symbol() is always subject to the relation (1) = a. [272] An integral expression may be regarded as a fraction whose denominator is unity: thus, a + b is the same as a+b, by the definition of quotient, (+). 1 = a + b. a is a rational fraction when a and b are integers. 5 8 5 8 The denominator of the fraction declares that the number of things in a certain group is 8, and the numerator 5 declares that of this group of 8 things 5 are taken; and in the use of the fraction the unit group is a group of 8 things. Thus, if one has a quantity of vinegar to measure, and finds that a gallon measure, which contains 8 pints, can be filled 13 times and that besides a pint measure can be filled 5 times, the measure of the quantity of vinegar is 13 gallons. A gallon is the unit, and the denominator 8 declares that the unit gallon is divided into 8 parts, and that 5 of these parts, or pints, are taken in the fraction Heregal. means one of the eight equal parts of a gallon (a pint), or briefly, 1 eighth of 5 a gallon; and gal. means 5 of the eight equal parts of a gallon (or 8 5 pints) or briefly, 5 eighths of a gallon; and similarly for any other fraction. 127. The rules of division are purely the formal consequences of the fundamental laws of the multiplication of numbers, III, IV, V. definition IX (261), theorem XI (63), and the corresponding laws of addition and subtraction. The rules of division, or, what is the same thing, the rules of the operation of fractions, can be deduced in the same way (872) as the rules of subtraction (38, 1-5). They follow without regard to the meaning of the symbols a, b, c, =, +, —, ab, ¦ (872). b 128. The rule governing the dependence of signs of a fraction upon the signs of its terms is deduced from the rules of the signs of products when the factors have different signs (341, 6 and 8), thus: +15 = = 3, for +15= (+15) (-5) = (-3) (-5) [241,8] 3, for 21= -21 = (+27) (+7)=( − 3) (+7) [841,6] -7 35 for -35= P) (—5)=(+7) (— 5). [?41,6] 5 Proofs: According to the rules used in establishing the preceding rule, we have: |