CHAPTER VIII. COMMON FACTORS AND MULTIPLES. Highest Common Factor. 137. Common Factors. A common factor of two or more integral numbers in Arithmetic is an integral number that divides each of them without a remainder. 138. A common factor of two or more integral and rational expressions in Algebra is an integral and rational expression that divides each of them without a remainder. Thus, 5 a is a common factor of 20 a and 25 a. 139. Two numbers in Arithmetic are said to be prime to each other when they have no common factor except 1. 140. Two expressions in Algebra are said to be prime to each other when they have no common factor except 1. 141. The greatest common factor of two or more integral numbers in Arithmetic is the greatest number that will divide each of them without a remainder. 142. The highest common factor of two or more integral and rational expressions in Algebra is an integral and rational expression of highest degree that will divide each of them without a remainder. Thus, 3 a2 is the highest common factor of 3 a2, 6 a3, and 12 a1; 5x22 is the highest common factor of 10 x3y2 and 15 x2y2. For brevity, we use H. C. F. for highest common factor. 1. Find the H. C. F. of 42 a3b2 and 30 a2b1. 42 a8b2 30 a2b4 = 2 × 3 × 5 × aa × bbbb. .. the H. C. F. = 2 × 3 × aa × bb = 6 a2b2. 4x2-4x-80; 2x2-18x+40; 2x2-24x + 70. 4x2-4x-80=4(x2-x-20) =4(x-5)(x+4); 2x2-18x+40=2(x2-9x+20) =2(x-5) (x-4); 2x2-24x+70 = 2 (x2 - 12x+35) =2(x-5) (x-7). ..the H.C.F. 2(x-5). Therefore, = 143. To Find the H. C. F. of Two or More Expressions, Resolve each expression into its prime factors. The product of all the common factors, each factor being taken the least number of times it occurs in any of the given expressions, is the highest common factor required. NOTE. The highest common factor in Algebra corresponds to the greatest common measure, or greatest common divisor, in Arithmetic. We cannot apply the terms greatest and least to algebraic expressions in which particular values have not been given to the letters contained in the expressions. Thus a is greater than a2, if a stands for; but a is of lower degree than a2. 8. (x − 1)2 (x + 2)2 and (x − 3) (x + 2)3. 9. 24 a2b3 (a + b) and 42 a3b (a + b)2. 10. x2 (x − 3)2 and x2 - 3x. 12. x2 - 4 x and x2 − 6 x + 8. 11. 2-16 and x2 + 4x. 13. x2-7x+12 and x2 - 16. 16. x2+3xy 10 y2 and x2 -2xy - 35 y2. 17. x4 - 2x3- 24x2 and 6x5 18. x3-3x2y and x3- 27 y3. 19. 164x3 and 1 - 6 x 180 x3. 4x + 16 x2. 23. 3a+15 ab 72 a262 and 6 a3 30 a2b+36 ab2. 24. 6 x2y — 12 xy2 + 6 y3 and 3 x2y2 + 9 xy3 — 12 ya. 25. 1 - 16 c4 and 1 + c2 – 12 c1. - 6x+1; 27 x3 — 1. 54; x2 x- ·42; x2 - 2 x — 48. 28. 8x+278; 4x2+12xy +9y2; 4x2-9 y2. 29. x3 — x2y — xy2 + y3; x2 — y2; x2 + 2xy + y2. Lowest Common Multiple. 144. Common Multiples. A common multiple of two or more integral numbers in Arithmetic is a number that is exactly divisible by each of the numbers. A common multiple of two or more integral and rational expressions in Algebra is an integral and rational expression that is exactly divisible by each of the expressions. 145. The least common multiple of two or more integral numbers in Arithmetic is the least integral number that is exactly divisible by each of the given numbers. The lowest common multiple of two or more integral and rational expressions in Algebra is an integral and rational expression of lowest degree that is exactly divisible by each of the given expressions. We use L. C. M. for lowest common multiple. 1. Find the L. C. M. of 42 a3b2; 30 a2b*; 66 ab3. 42 a3b2 = 2 x 3 x 7 x a8 x b2; 30 a2b1 = 2 × 3 × 5 × a2 × b1; The L. C. M. must evidently contain each factor the greatest number of times that it occurs in any expression. .. the L. C. M. = 2 × 3 × 7 × 5 × 11 × as x b4 = 2310 a3b1. 2. Find the L. C. M. of 4x2 - 4x 80 and 2 x2 - 18x + 40. 4x2-4x-804 (x2x20) = 4(x-5) (x+4); 2x2-18x+40=2(x2- 9x+20)=2(x-5) (x-4). .. the L. C. M. 4 (x − 5) (x + 4) (x-4). Hence, 146. To Find the L. C. M. of Two or More Expressions, Resolve each expression into its prime factors. The product of all the different factors, each factor being taken the greatest number of times it occurs in any of the given expressions, is the lowest common multiple required. 17. (a+b)2; (a — b)2; a2 — b2. 18. (a+2c); (a-2c)2; a2 - 4 c2. 19. 4 xy (x + y)2; 2x2 (x2 — y2); x3 (x+y). 20. x2+7x+12; x2+6x+8; x2+5x+6. 21. 1-y2; 1-y3; 1+y. x2 — y2; x2 - 2xy + y2. 22. x2+2xy + y2; 24. y2 — 1; y3 + y2 + y + 1; y3 − y2 + y − 1. 25. (x + y)2 - 22; (x + y + z)2; x + y − z. 26. x2 - (a + b) x + ab ; x2 − (a + c) x + ac. 27. x2 + 3 xy + 2 y2; x2 + 5 xy +4y2; x2-6xy — 7 y2. 28. x2 - 7 xy + 12 y2; x2 — 6 xy + 8 y2; x2 — 5 xy + 6 y2. |