CHAPTER VIII HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE PART I. HIGHEST COMMON FACTOR 79. Common Factors. In arithmetic a factor of each of two or more numbers is called a common factor of the numbers. Thus, 3 is a common factor of 9 and 15; 5 is a common factor of 10, 15, and 25. In the same way, we say in algebra that x is a common factor of 2 x and 5 x; rs is a common factor of 2 r252 and rs; a+b is a common factor of (a+b)2 and a(a+b). 80. Prime Factors. A number that has no factor except itself and unity is called in arithmetic a, prime number. Such a number when used as a factor is called a prime factor. Thus, the prime factors of 10 are 5 and 2; the prime factors of 36 are 3, 3, 2, and 2. In the same way, we say in algebra that the prime factors of 3 abc are 3, a, b, and c; the prime factors of aPb(a? — 62) are a, a, b, a,b, and a+b. 81. To Find Common Factors. As soon as we factor each of several numbers into their prime factors, we can easily pick out the common factors. For example, to find the common factors of 54, 90, 108, and 180, we have 54=3 · 3 · 2 · 3, . 90=3 · 3 · 2 · 5, 108=3 · 3 · 2 · 3 · 2, 180=3 · 3 · 2 · 2 · 5. The common factors are therefore 3, 3, and 2. The same process is followed in algebra. Thus, in finding the common factors of abc, ab, ab?, and 3 ab, we write abc=a ·b·c, 3 ab=3 ·a·b. 82. Highest Common Factor (H. C. F.). The product of all the common prime factors of two or more numbers or expressions is called their highest common factor. It is called the highest common factor because it contains all the common factors. The abbreviation for it is H. C. F. For example, to find the H. C. F. of 90, 60, and 120, we write 90=3 · 2.5.3, 120=3 · 2:5 · 2 · 2. Since 3, 2, and 5 are the only common factors, their product, 30, is the H. C. F. We find the H. C. F. of algebraic expressions in the same way, as is illustrated in the following examples. EXAMPLE 1. Find the H. C: F. of 6 a363, 2 ab2, and 8 a3b2. SOLUTION. 6a3b3=3 · 2 · a: a·a·b·b·b, 2 ab2=2·a·b·b, 8 a3b2=2 · 2 · 2 ·a·a·a·b· b. Picking out all the factors common to the three expressions, we see VIII, $ 83] · HIGHEST COMMON FACTOR UICHEST COMMON FACTOR. 133 that they are 2, a, b, and b. Hence the H. C. F. is 2. a·b·b, or 2 abz. Ans. EXAMPLE 2. Find the H. C. F. of a2–4 a+4, a? — 2 a, and a2–6 a+8. Solution. a?–4 a+4=(a-2)(a–2), (Formula VI) a2—2 arala-2), (Formula IV) The only common factor being a-2, the H. C. F. is a-2. Ans. EXAMPLE 3. Find the H. C. F. of 3 x2 +3 x – 18, 6 x2+36 x+54, and 9 x2–81. SOLUTION. 3 x+3 x-18=3(x2+x-6)=3(x+3)(x-2), 6 x2+36 x+54=6(x2+6 x+9)=2 · 3(x+3)(x+3), 9x2—81=9(x2—9)=3 · 3(x+3)(x-3). The common factors being 3 and (x+3), the H. C. F. is 3(3+3), or 3 x+9. Ans. 83. Important Property of the H. C.F. Since the H. C. F. of several expressions is always made up of the factors common to them all, it is an exact divisor of each of the expressions. Thus, the H. C. F. of 5 a2b3, 3 a262(c+d), 4 a3b3 is a2b2. Observe that this is contained in the first expression 5 b times, in the second expression 3(c+d) times, and in the third expression 4 ab times. For this reason, the highest common factor is called in arithmetic the greatest common divisor, and is represented by the letters G. C. D. ORAL EXERCISES State the H. C. F. of the expressions in each of the following exercises. 1. 12 and 18. 5. a-b3 and aba. 2. 16 and 24. 6. x2y223 and xy_22. 3. xay and xy. 7. rösy, rasy, and rsay. 4. a4b4 and a368. 8. 2 mn, 3 mn?, and 6 mnp. NOTE. The H. C. F. of several monomials is most easily found by picking out the lowest power of each letter and multiplying them together. Thus, in finding the H. C. F. of x2y3z2, x4y423 and x3y52, the lowest power of x is x?, the lowest power of y is y; and the lowest power of 2 is z. Therefore, the H. C. F. is 22 • y• z, or simply xłysz. Ans. By use of this Note state the H. C. F. in each of the following exercises. 9. xạyoz, x3y3z2, and xy223. 12. 2 f?g2h, goh27, and 3 hizj. 10. manoq, mnaq and mn. 13. 4 xoy and 8 yuz. 11. pq3pts and p3q2r. 14. axy and bx_y_2. State the H. C. F. in each of the following exercises (use Formulas of $ 78). 15. a? — 62 and a? — 2 ab +62. 18. q2+7 a+12 and q? – 9. 16. a2 + ab and a2 +2 ab +62. * 19. a3 — 33 and a-b. 17. a?+2 a+1 and a2–1. * 20. a3 +63 and a2+2 ab+b?. 84. General Rule for H. C. F. From what we have seen about the H. C. F. we may state the following general rule. RULE FOR FINDING THE H. C. F. OF TWO OR MORE EXPRESSIONS. Resolve each expression into its simplest factors. Find the product of all the common factors, taking each factor the least number of times it occurs in any of the given expressions. This product is the required H.C. F. of the given expressions. WRITTEN EXERCISES Find the H. C. F. of the expressions in each of the following exercises. Check your answer by showing that it is contained exactly in each of the given expressions. (See $ 83.) 1. x2 - y2 and x2 – 2 xy+y?. 4. 22—9 and x2 — 2–6. 5. p2— 2 r, p2—r—2, and p2—3 r+2. 6. y4+3 y3+2 y, yu+y?, and y4+7 y3 +6 ye. 7. a?-1, aż— 2 a+1, and a? — 13 a+12. 8. 1-5x, 1-10 x+25 x?, and 1-25 xạ. 9. 3 62—33 b and 62 – 7 6-44. 10. 3 yö — y, 3 y: — 6 y2 +3 y, and 6 y3+12 y2-15 y. 11. a-b, (a−b)?, and (a−b). 12. a2+2 a 15 and a2–4 a +3. 13. 2 a2 +4 a and 4 a'+12 a+8 a. 14. 24 — x?y2 and x2y+xy. 15. at - a3 – 2 a’, a4 — 2 a'— 3 a, and at — 3 a3 — 4 a. 16. Q4 — 2 a2+1 and a? — 2 a+1. 17. a2b-63, ab+b, and a262 — 64. 18. 22— 25, 22—7r+10, and r2 —r—20. 19. x2-(y+1)?, y2– (x+1)?, and 1–(x+y)2. 20. x2 — y2 and x3+x2y + xy2 +y3. [HINT. To factor the second expression, see § 54.] 21. 56+20 and 5 ab+20 a – 26 – 8. 22. y2 – 11 y+30 and yz-5 z+y2 – 5 y. 23. 3 p5+9 g4 – 3 ry, 5 r292 +15 rs2–5 sa, and 7 ar2 +21 ar – 7 a. 24. 363–3 6,3 63 – 6 62+3 b, and 6 63+12 62 – 18 b. 25. (1-x)?, x2 – 1, and x2 – 2 x+1. * 26. 03 — 53, a? — B2, and a–b. * 27. a3 + b3 and (a+b)2. * 28. 2+1 and x2 — x+1. * 29. (m+n)?, m3+no, and m2 — n2. * 30. 23—1, x2– 10 x+9, and x2 — X. * * * * * |