If the two given quantities be prime to each other, the least common multiple is equal to their product. If one of the two quantities be a multiple of the other, the former is their least common multiple. The least common multiple of three quantities is found by first finding the least common multiple of two of them, and then the least common multiple of this and the third quantity. And similarly of four and any number of quantities. It may also be noted that the highest common divisor of any number of algebraical quantities is equal to the least common multiple of all the common divisors of the quantities. And the least common multiple of any number of algebraical quantities is equal to the highest common divisor of all their common multiples.* 4. Prop. If one quantity measure the product of two others, but is prime to one of them, it measures the other. Let c measure ab, but be prime to a, then c measures b. For let a be greater than c. Then let c be contained in a, x times with a remainder r, Again, let r be contained y times in c with a remainder 8, It is manifest that also is less than r; and as the remainder is continually diminishing at each step, it will at last become unity. For if there be any other remainder, the division may be continued, and if there be no remainder, then the last divisor will measure a and c; and therefore a and c have a common measure which is contrary to the hypothesis. Suppose then that after the third division the remainder is 1, Now c measures ab and ber, it .. measures ab-bex or br, and .. it measures bc-bry or bs, and .. its measures br-bsz or b. Hence.. if c measure ab, and do not measure a, it measures b. The reasoning is similar if c be supposed greater than a. To find the least common multiple of 2x2+9xy+9y2 and 3x2+4xy-15y3. The highest common divisor of these expressions is x+3y. .. Least common multiple = (3x2+4xy-15y2)(2x2+9xy+9y2) = (3x2+4xy — 15y2)(2x+3y) · 6x 3 +17 x2 y — 18xy2 — 45y3. EXERCISES. I. 1. Explain the process of finding the highest common divisor of two algebraical expressions, and shew that the result is not affected: (1) by removing any factor which is common to every term of a divisor at any stage of the operation which is not found in every term of the corresponding dividend; (2) by introducing into any dividend a factor which is not contained in every term of the corresponding divisor. 2. Explain in what respects the process of finding the highest common divisor of two algebraical polynomials differs from that of finding the greatest common measure of two numbers. 3. If the quantity D measure A and B ; it also measures mA±nB. 4. Does the common process for finding the highest common divisor of two algebraical expressions enable us to determine all the common divisors whenever they exist? 5. Every other multiple of any two quantities is the same multiple of their least common multiple. 6. If m be the least common multiple of a and b, every other common multiple of a and b is a multiple of m. 7. If m be the least common multiple of a and b, m' the least common multiple of m and c, and m" the least common multiple of m' and d; shew that m" will be the least common multiple of a, b, c, d. 8. Find two numbers whose greatest common measure is 2 and whose least common multiple is 36. 9. Determine that algebraic expression which involves the lowest possible powers of x that can be exactly divided by x2+5x+6 and x2+7x+8. 10. Shew that the highest common divisor of two compound algebraical quantities is the least common multiple of all the common divisors. II. Find the highest common divisor of the following quantities: 1. 3x-2x-1 and 6x-x-1. 2. x-3x+2 and +2x-3. 3. 3x-2x-2x+1 and x-x-3x+4x-1. 4. 6x'y+xy-xy and 4x-6x'y-4xy+3y3. 5. 11x-9ax3-a2x2-a1 and 13x-10ax3-2a2x2-a1. 6. 2x-11x2-9 and 4x+11x+81. 7. 3x-3x-53x3-43x2+34x+30 and 3x+3x-53x+43x2+34x-30. 8. 3x-10x+15x+8 and x-2x-6x+4x2+13x+6. 9. x+x3-2x2+x-1 and x-2.x1+3x3-3x2+2x-1. 10. x+6x+11x+6, x2+10x2+31x+30, and 3+10x+23x+14. 11. Ão—1 and æo—1+x(x1−1)+x2 (x2−1). 12. 3x2-(4a+2b)x+2ab+a2 and x3—(2a+b)x2+(2ab+a2)x−c©b. 16. x2+(a+b+c) x2+(be+ac+ab)x+ab+ab2 and +(a+2b)x2+(ab+ac+b2)x+a2e+abc. 17. a3+b3+c3-3abc and a(a+2b)+b(b+2c)+c(c+2a). 18. nx+1(n+1)x" +1 and x"—nx+n−1. III. Find the least common multiple of the following quantities: 1. na"-1x, (n-1)a"-2x2, (n—2)a"-x3, and (n-4)a*—*x*. 2. 3x2-5x+2, and 4x3-11x+x+6. 3. x3—a2x—ax2+a3, x1—a1 and ax3+a3x—a2x2 — a1. 4. x-2x-6x3+4x2+13x+6, and 3x+4x3-6x-12x-5. 5. a3+b3+3ab(a3 +b3)+a2b2(a+b), and 2a+263+5ab(a3+b3)+6a2b2(a+b). 6. a3-2 and a2-x2. 7. x2-4a3, x3+2ax2+4a2x+8a3, and 2-2ax2+4a3x-8a3. 8. 21x2-13x+2, 28x2-15x+2, and 12x2-7x+1. 9. x3—y3, x3+y3, x1+x2y2+y*. 10. x3+6x2+11x+6, x3+7x3+14x+8, x3+8x2+19x+12, and +9x+26x+24. 11. 6(a3—b3)(a—b)3, 9(a1—b1) (a—b), and 12(a2-b2)3. 12. x, x+1, x2+1, x3+1, and +1. 13. x, x-1, -1, x3-1, and 2-1. 14. 6x3—13x+6, 12x2—5x−2, and 15x2+2x−8. 15. 2-1, x-1 and x-1. IV. 1. Find the value of a for which the following fraction admits of x-ax2+19x-a-4 reduction : x3—(a+1)x2+23x—a—7 2. Find the greatest common divisor of +y+z3 and 25+y+25 when x+y+x=0. 3. If aTM+am-1b+b” and aTM+abTM-1+b have a common divisor, m being an integer, prove that it is of the form 3p-1, and find the greatest common measure. 4. If the greatest common measure of a, b, c be an odd number m; then the greatest common measure of a+b, b+c, c+a will be 2m, if a, b, c be odd numbers, or m if they are not. 5. If p, q, r be the successive quotients in finding the highest common divisor of the numerator and denominator of the fraction 6. Shew that ax+bx+c and a+b+cx have a common quadratic factor, if b22= (c2 —a2+b2) (c2—a2+ab); 7. The expressions ax2+bx+c and ax2+mbx+mic have a common divisor, if (m+1)*ac = mb3. 8. If D be the greatest common measure, and M the least common multiple of two decimals which have m and m+n decimal places re spectively; prove that is a multiple of 10". D 9. If the least common multiple of ax3+cx+d and a' x3+cx2+d' be an expression of the fourth degree; shew that a3d=aa' c+a'3. 10. Shew that if x+c be the highest common divisor of x2+ax+b and x2+mx+a; their least common multiple is 3+(a+m—c)x2+ (am—c2)x+(a−c)(m—c)c. 11. If ax3+bx+c and mx3+nx+p have a common factor of the form x+q; shew that (pa-mc)3 = (nc—pb)(na—mb)2. 12. Find the highest common divisor of "-1) (x*+1+1) and (-1)(x+1+1). 13. Find the highest divisor of "-y" and "-y", (1) when m and n are of the forms 4p, 4p+2, and (2) of the forms 4p+1, 4p+3. 14. Explain how the highest common divisor of two algebraical expressions, does not always give the greatest common measure of the numbers which result, when particular numerical values are given to the algebraical symbols. Exemplify in finding the highest common divisor of x-x2-2x-1 and x+2x+x2-1; and of 3x2+ax-4a2 and 6x3-7ax2-20a3x; and their greatest common measures, when x=4 and a = 1. 15. If M be the least common multiple of the three numbers. ab+ac+bc a, b, c, shew what are the conditions that M may be integral... 16. m, m, m2, m, and d, d1, d2, d be the least common multiples and the highest common divisors of A, B, C; of B, C; of C, A; and of A, B, respectively; Shew that mmm ARC and ddd, _ ABC m d d m 17. If 1, 2, 3 are the lowest common multiples of B and C, of C and A, of A and B, respectively; if 91, 92, 9s are the highest common divisors of the same pairs; and if L, G are the lowest common mul-tiple and highest common divisor of A, B, and C; prove that FRACTIONS. Art. 1. Algebraic fractions are subject to the same principles as arithmetical fractions, and every fraction may be considered as representing the quotient arising from the division of the numerator by the denominator. The fraction a cannot be called either a proper or improper fraction, while the values of the numerator and denominator are not assigned; but it may be considered a proper or improper fraction according as the numerator is less or greater in form than that of the ma+b denominator, as ma+b a is a proper and is an improper fraction. a Every quantity having an integral form can be expressed as a fraction, by placing unity for the denominator, as a is the same as a 2. The general properties of algebraic fractions are the same as those of numerical fractions, with this distinction, that general symbols are employed in the former and definite numbers in the latter. a A fraction is multiplied by any quantity by multiplying the numerator by that quantity; as mat is the product of the fraction by the quantity m. b Conversely. A fraction is divided by any quantity m by dividing the numerator by that quantity, as is the quotient of ma divided by m. a ī b A fraction is also divided by any quantity by multiplying the denominator by that quantity, as is the quotient of the fraction divided by the quantity m. a mb a b Conversely. A fraction is multiplied by any quantity by dividing the denominator by that quantity, as is the product of the fraction a b Hence, a fraction is multiplied by any integral quantity either by multiplying the numerator or dividing the denominator by that quantity. And, a fraction is divided by any integral quantity, either It must always be remembered that the value of a fraction depends on the relative values of the numerator and denominator, and not on their absolute values. + If m=b, then 3×6=a; that is: if a fraction be multiplied by a quantity equal to its denominator, the product is equal to the numerator. |