then, since m and n have no common measure but unity (otherwise D would not be the Greatest Common Measure of A and B), their Least Common Multiple evidently = mn; therefore L. C. M. of mD and nD=mnD Hence this Rule ; =(mDxnD)÷ by D, =(A× B)÷D, =product of the expressions÷their Greatest Common Measure. "Divide the product of the proposed expressions by their Greatest Common Measure." In practice it is better to divide one of the quantities by the Greatest Common Measure and multiply the other by the quotient. Note. The Least Common Multiple of monomials, and of other quantities which can easily be expressed as monomials, may be found by inspection. 77. Every common multiple of A and B is a multiple of their Least Common Multiple (M). For let (m) be some other multiple of A and B. Now if (m) be not a multiple of M, let it contain Mr times with a remainder 8, which of course will be less than M: or let Now since A and B both measure M, they measure rM, Art. (71), and therefore m-rM or s; but s is less than M, or M is not the Least Common Multiple. Hence m will contain M exactly, i. e. without a remainder, and it is therefore a multiple of M. Ex. XXXV. Examples in L. C. M. worked out. Ex. 1. Find the L. C. M. of 6a2 and 9a1. The G. C. M. of 6a2 and 9a1 is evidently 3a2, Ex. 2. Find the L. C. M. of 12x2y3z1 and 8y31⁄23x2. The G. c. M. of 12x2y3z1 and 8y31⁄23x2 is evidently 4x2y3ï3, Ex. 3. Find the L. C. M. of x2-a2 and (x-a)2. x2-a2=(x+α) (x-a), and (x-a)2= (x-a) (x− a), x-a is the G. C. M. of 2-a2 and (x-a)3, a3-x3=(a-x) (a2+ax+x2), and a2x2= (a−x) (a + x), Ex. 5. Find the L. C. M. of (a2x-ax2)2 and ax (a2- x2)2. (a x-ax2)2={ax (a-x)=a2x2 (a-x) (α-x), ax (a2 — x2)2= ax (a+x)2 (a−x)2, G. C. M.=ax (a− x)2. .. L. C. M. a2x2 (a−x) (a− x) (a + x)2 = = a2x2 (a3 - x2)2. Ex. XXXV. Examples for Practice. Find the L. C. M. of 1. a2 and 3ab; 8a3 and 12a1; 9ax3 and 24x3. 2. 10ab and 15bc; 15æ1y2 and 25x2y^; 33x1y and 3xy1. 3. ax+ay and ax-ay; a-x and a2—x2. 4. 3(a+x) and 4 (a2 — x2); a2bx — ab3y and abx+b3y. 5. x2+xy and (x+y)2; b2 and b(b+ax). 6. 6x2+5x-6 and 6x2-13x+6; x2-1 and x2+4x+3. 7. x3-x2y and x2-y2; 12x2-17ax+6a2 and 9x2+6ax-8a2. 8. x2-I and x3+1; 6x2+13x+6 and 8x2 + 6x −9. 9. x3-7x2+6x and x2+2x-3; and (x+1)2 and ∞3 +1. 10. x3+bx2+ax + ab and x2 - (a—b) x — ab. - 11. 15a+10a4b+4a3b2+6a2b3-3ab4 and 12a3b2+38a2b3+16ab1-10b3. 12. x3+x1y+x3y2+x2y3 +xy1+y3 and x3 — x3y+x3y2 — x2y3 + xy3 — y3. 78. To find the Least Common Multiple of three Algebraical expressions. Let A, B, and C be the expressions; find M, the Least Common Multiple of A and B, by Art. (76); then the Least Common Multiple of M and C will be the Least Common Multiple of A, B, and C. For every multiple of M is a multiple of A and B, therefore every common multiple of M and C is a common multiple of A, B, and C ; and every multiple of A and B is a multiple of M, and therefore every multiple of A, B, and C is a multiple of M and C; therefore the multiples of A, B, and C are the same as those of M and C, and therefore the Least Common Multiple of M and C is the Least Common Multiple of A, B, and C. This Rule may be extended to any number of quantities. Ex. XXXVI. Examples worked out. Ex. 1. Find the L. C. M. of x-1, x2-1, and x2-2x+1. .. L. C. M. of x-1 and x2-1=(x+1)(x−1). Now to find the L. c. M. of (x+1)(x-1) and x2 -2x+1 . L. C. M. of (x+1)(x-1) and x2-2x+1=(x+1)(x−1)(x−1) Ex. 2. Find the L. c. M. of x2-y2, (x − y)2, and ∞3 – y3. x2-y2= (x + y) (x − y) (x-y)=(x-y) (x-y), .. L. C. M. of x-y and (x-y)=(x+y) (x − y) (x-y). .. L. C. M. of x3 —y3 and (x+y) (x−y) (x− y) 12. x(x+1)(x+2), x (x-1) (x-2), and x(x+1)(x-1). 13. x, α, x(x+a), and a (x− a). 14. 2x+2, x+2, 2 (x+3), and 3 (x+2) (x+3). 15. (a-b) (a-c) (x+a), (a−b) (b-c) (x+b), and (a–c) (b−c) (x+c). 16. a+b, 3 (a−b), 2 (a2 — b2), 6 (a+b)2, and a2 + b2. 17. x3-3x2+3x-1, x3 — x2-x+1, ∞1 −2,x3 +2x-1, 18. x2-y2, x2 + y2, (x − y)2, (x+y)2, x3 —y3, and x3 + y3. 19. (1+x)3, (1+∞)3, and (1+∞)3. 20. (a-b+c), (a+b−c), (c-a+b), c-(a-b), and a2-(b-c). ALGEBRAICAL FRACTIONS. 79. The Rules for the treatment of Algebraical Fractions are founded on the same principles as those in Arithmetical Fractions. The Rules which in Arithmetical Fractions were shewn to be true in particular examples, will now be proved generally. a 80. The fraction denotes that the unit has been divided into b b equal parts, a of which parts are taken. a b The fraction also denotes the quotient of a by b; since it is clearly the same thing, whether we divide the unit into b equal parts and take a of such parts, or taketh part of a units, i.e. divide a by b. 81. Every integral quantity may be considered as a fraction whose denominator is 1; thus a= a 82. If the numerator and denominator of a fraction be both multiplied, or both divided by the same quantity, its value is not altered. a For in the fraction if we divide the unit into mb instead of b equal parts, each of these parts is th of the original part, and m times as m many parts must be taken, to make up an equal fraction; that is 83. If the signs of all the terms both in the numerator and denomi nator of a fraction be changed, its value will not be altered. For + ab |