119. The following statement gives the best practical notion of what is meant by the term least common multiple in Algebra, as it shews the sense of the word least here. When the least common multiple of two or more expressions is divided by those expressions the quotients have no common measure. Take the first example of Art. 118, and divide the L.C.M. by the expressions; the quotients are 5b2d and 4ac, and these quotients have no common measure. Again; take the second example of Art. 118, and divide the L.C.M. by the expressions; the quotients are 6a2cy3, 4b2c2x3yz3, and 3ab3x3z3, and these quotients have no com mon measure. 120. The notion which is supplied by the preceding Article, with the aid of the Chapter on Factors, will enable the student to determine in many cases the L. C.M. of compound expressions. For example, required the L. C. M. of 4a2 (a+b)2 and 6ab (a2 — b2). The L.C. M. of 4a2 and 6ab is 12a2b. Also (a+b) and a2-b2 have the common factor a+b, so that (a+b) (a+b) (a−b) is a multiple of (a+b)2 and of a2-b2; and on dividing this by (a+b)2 and a2-b2 we obtain the quotients a-b and a+b, which have no common measure. Thus we obtain 12a2b (a+b)2 (a−b) as the required L. C. M. 121. The following may be given as the definition of the L.C.M. of two or more compound expressions. Let two or more compound expressions contain powers of some common letter; then the expression of lowest dimensions in that letter which is measured by each of these expressions is called their least common multiple. 122. We shall now shew how to find the L. C. M. of two compound expressions. The demonstration however will not be fully understood at the present stage of the student's knowledge. Let A and B denote the two expressions, and D their greatest common measure. Suppose A=aD, and B=bD. Then from the nature of the greatest common measure, a and b have no common factor, and therefore their least common multiple is ab. Hence the expression of lowest dimensions which is measured by aD and bD is abD. And abD=Ab= Ba= AB D' Hence we have the following Rule for finding the L. C. M. of two compound expressions. Divide the product of the expressions by their G. C. M. Or we may give the rule thus. Divide one of the expressions by their G. C. M., and multiply the quotient by the other expression. 123. For example; required the L. C. M. of x2-4x+3 and 4x3-9x2-15x+18. The G. C. M. is x-3; see Art. 105. Divide x2-4x+3 by x-3; the quotient is x-1. Therefore the L.C.M. is (x-1)(4x3-9x-15x+18); and this gives, by multiplying out, 4x4-1323 — 6x2+33x-18. It is however often convenient to have the L. C. M. expressed in factors, rather than multiplied out. We know that the G. C. M., which is x-3, will measure the expression 43-9x2-15x+18; by division we obtain the quotient. Hence the L. C. M. is (x − 3) (x − 1) (4x2 — x − 6). 124. It is obvious that, every multiple of a common multiple of two or more expressions is a common multiple of those expressions. 125. Every common multiple of two expressions is a multiple of their least common multiple. Let A and B denote the two expressions, M their L. C. M.; and let N denote any other common multiple. Suppose, if possible, that when N is divided by M there is a remainder R; let q denote the quotient. Thus R=N-qM. Now A and B measure M and N, and therefore they measure R (Art. 106). But by the nature of division R is of lower dimensions than M; and thus there is a common multiple of A and B which is of lower dimensions than their L. C.M. This is absurd. Therefore there can be no remainder R; that is, N is a multiple of M. 126. Suppose now that we require the L. C. M. of three compound expressions, A, B, C. Find the L. C. M. of any two of them, say of A and B; let M denote this L. C. M.; then the L. C. M. of M and C will be the required L. C.M. of A, B, and C. For every common multiple of M and C is a common multiple of A, B, and C, by Art. 124. And every common multiple of A and B is a multiple of M, by Art. 125; hence every common multiple of M and C is a common multiple of A, B, and C. Therefore the L. C. M. of M and C is the L. C. M. of A, B, and C. 127. In a similar manner we may find the L. C. M. of four expressions. 128. The theories of the greatest common measure and of the least common multiple are not necessary for the subsequent chapters of the present work, and any difficulties which the student may find in them may be postponed until he has read the Theory of Equations. The examples however attached to the preceding chapter and to the present chapter should be carefully worked, on account of the exercise which they afford in all the fundamental processes of Algebra. EXAMPLES. XIII. Find the least common multiple in the following examples. 1. 4a2b, 6ab2. 2. 12a3b2c, 18ab2c3. 3. Sa2x2y3, 12b2x3y2. 4. (a-b)2, a2 — b2. 5. 4a (a+b), 6b(a3+b3). 6. a2-b2, a3-b3. 7. x2-3x-4, 2-x-12. 8. x3+5x2+7x+2, x2+6x+8. 9. 12x2+5x-3, 6x3+x2-X. 10. x3-6x2+11x-6, x3-9x2 + 26x-24. 11. 23-7x-6, x3+8x2+17x+10. 12. 24+23+2.x2 + ~ + 1, 24-1. 13. xa - 2x3-3x2+8x-4, x1-5x3+20x−16, 14. x2+a2x2+a1, x2-ax3-a3x+aa. 15. 4a3b2c, 6ab3c2, 18abc3. 16. 8(a2-b), 12 (a+b), 20 (a-b). 17. 4 (a+b), 6 (a2-b2), 8 (a3+b3). 18. 15(a2b-ab2), 21 (a3 — ab2), 35 (ab2+b3). 20. 2-1, 2+1, 2+1, 28–1. 21. x2-1, x3+1, x3−1, No6+1. 22. x2+3x+2, x2+4x+3, x2+5x+6. 23. x2+2x+3, x3+3x2-x-3, x3+4x2+x−6. XIV. Fractions. 129. In this Chapter and the following four Chapters we shall treat of Fractions; and the student will find that the rules and demonstrations closely resemble those with which he is already familiar in Arithmetic. a b a b 130. By the expression we indicate that a unit is to be divided into b equal parts, and that a of such parts are to be taken. Here is called a fraction; a is called the numerator, and b is called the denominator. Thus the denominator indicates into how many equal parts the unit is to be divided, and the numerator indicates how many of those parts are to be taken. Every integer or integral expression may be considered as a fraction with unity for its denominator; that is, for b+c 1 3 131. In Algebra, as in Arithmetic, it is usual to give the following Rule for expressing a fraction as a mixed quantity. Divide the numerator by the denominator, as far as possible, and annex to the quotient a fraction having the remainder for numerator, and the divisor for denominator. The student is recommended to pay particular attention to the last step; it is really an example of the use of brackets, namely, +(−x+2)= − (x − 2). 132. Rule for multiplying a fraction by an integer. Either multiply the numerator by that integer, or divide the denominator by that integer. a Let denote any fraction, and e any integer; then b willxc. For in each of the fractions and the b ac a ас unit is divided into b equal parts, and c times as many a a a then will XC= For in each of the fractions bc bc |