COMMON FACTORS. WRITTEN WORK. 168. ILLUSTRATIVE EXAMPLE I. What numbers are factors of both 18 and 24 ? Explanation. — Separating 18 and 24 into their prime factors, we find 2 and 3, and conse18 = 2 x 3 x 3 quently 6 (which is the product of 2 and 3), to 24 -- 2 x 2 x 2 x 3 be factors of both 18 and 24. Ans. 2, 3, and 6. Name any common factor of 12 and 15; of 12 and 18; of 30 and 40. 169. Numbers that have no common factors are said to be prime to each other. Thus, 14 and 15 are prime to each other, though they are not prime numbers. 170. The greatest factor which is common to two or more numbers is their greatest common factor. What is the greatest factor which is common to 18 and 24 ? to 40 and 50 ? to 45 and 54 ? 171. We have seen that 6, the greatest common factor of 18 and 24, is the product of 2 and 3, the only prime factors common to 18 and 24. The greatest common factor of any two or more numbers is the product of all the prime factors which are common to those numbers. NOTE. The letters g. c. f. are used for greatest common factor. WRITTEN WORK. To find the Greatest Common Factor. 172. ILLUSTRATIVE EXAMPLE II. Find the greatest common factor of 12, 30, and 48. Explanation. The prime factors of 12 are 2, 12 = 2 x 2 x 3 2, and 3. The product of such of these as are 6 common to 30 and 48 must be the g. c. f. re quired. We find that 2 is a factor of both 30 and 48 ; therefore 2 is a factor of the g. c. f. We find that but one 2 is a factor of 30; therefore only g. c. f. 2 x 3 one 2 is used as a factor of the g. c. f. We find that 3 is a factor of both 30 and 48 ; therefore 3 is a factor of the g. c. f. Thus the g. c. f. sought is 2 x 3, equal to 6. Hence the following Rule. 173. To find the greatest common factor of two or more numbers : Separate one of the numbers into its prime factors, and find the product of such of them as are common to the other numbers. 174. Examples for the Slate. Find the greatest common factor 47. Of 48, 56, and 60. 48. Of 24, 42, and 54. 49. Of 108, 45, 18, and 63. 50. Of 18, 36, 12, 48, and 42. NOTE. In Example 50, 18 is a factor of 36, and 12 of 48. The g. c. f. of 18 and 12 must be the g. c. f. of 18, 12, and their multiples 36 and 48 ; hence we need only find the g. c. f. of 18, 12, and 42. Find the greatest common factor 51. Of 42, 28, and 84. 53. Of 32, 18, 108, and 25. 52. Of 26, 52, and 65. 54. Of 114, 102, 78, and 66. 55. What is the width of the widest carpeting that will exactly fit either of two halls, 45 feet and 33 feet wide, respectively? 56. A has a piece of ground 90 feet long and 42 feet wide. What is the length of the longest rails that will exactly suit both its length and its width ? 57. What is the length of the longest stepping-stones that will exactly fit across each of three streets, 72, 51, and 87 feet wide, respectively? 58. What is the length of the longest curb-stones that will exactly fit each of four strips of sidewalk, the first being 273 feet long, the second 294, the third 567, the fourth 651 ? WRITTEN WORK. 175. When numbers cannot readily be separated into their factors, the following method for finding the greatest common factor may be adopted. ILLUSTRATIVE EXAMPLE. Find the greatest common factor of 52 and 91. Divide the greater number by the less, 52) 91 1 and then divide the less number by the 52 remainder, if there be any. Continue dividing the last divisor by the last re39) 52 (1 mainder until nothing remains. The last 39 divisor will be the g. c. f. sought. 13) 39 (3 NOTE. As the explanation of this method is some39 what difficult for younger pupils, it is not given here, but will be found in the Appendix, page 804. To find the g. c. f. of more than two numbers, find the g. c. f. of any two of them and then of that common factor and a third number, and so on till all the numbers are taken. 176. Find the greatest common factor 59. Of 323 and 663. 61. Of 6581 and 1127. 60. Of 147 and 966. 62.. Of 187, 442, and 969. For other examples in factoring, see page 123. MULTIPLES. 177. Name some numbers which are made by using 3 as a factor. Ans. 3, 6, 9, 12, etc. Any number made by using another number as a factor is a multiple of the number thus used. 178. Name the multiples of 4 and of 6 to 36. 4 . 6 , Which of these numbers are multiples of both 4 and 6? Numbers which are multiples of two or more numbers are common multiples of these numbers. Ans. {Multiples of A are 4, 8, 12, 16 , 20, 24, 28, 32, 36. Thus 12, 24, and 36 are common multiples of 4 and 6. 179. Oral Exercises. Name any six multiples of 5. Name three multiples of 12. Name all the multiples of 11 up to 140. Name any common multiple of 10 and 6. Of 3, 6, and 5. Least Common Multiple. 180. Name the least number which is a multiple of both 4 and 6. Ans. 12. The least number which is a multiple of two or more numbers, is the least common multiple of those numbers. Name the least common multiple of 2 and 5; of 6 and 9. 181. As any number contains all its prime factors, a multiple of any number must contain all the prime factors of that number. A common multiple of two or more numbers must contain all the prime factors of those numbers, and The least common multiple of two or more numbers is the least number which contains all the prime factors of those numbers. 182. ILLUSTRATIVE EXAMPLE I. What is the least common multiple of 6, 9, and 15? Explanation. — The least multiple of 6= 2 * 3 6 is 6, which may be expressed in the 9= 3x3 form 2 x 3. 15 = 3 x 5 The least multiple of 9 is 9, which may be expressed in the form 3 x 3. But in 2 x 3 x 3 x 5 = 90 6 we have already one of the factors (3) of 9; hence if we put with the prime factors of 6 the remaining factor (3) of 9, we shall have 2 x 3 x 3, which are all the factors necessary to produce the l. c. m. of 6 and 9. WRITTEN WORK. 1. c. m. The least multiple of 15 is 15, which may be expressed in the form 3x5. In the l. c. m. of 6 and 9 we have one of the prime factors (3) of 15; hence if we put t with the prime factors of 6 and 9 the remaining factor (5) of 15, we shall have 2 x 3 x 3 x 5, which are all the prime factors necessary to produce the l. c. m. of 6, 9, and 15. The product of these factors is 90, which is the l. c. m. sought. NOTE. In finding the least common multiple, the factors of the given numbers seldom need to be expressed, and the written work may be greatly reduced. Thus, in this example the written work may be simply 1. c. m. = 2 x 3 x 3 x 5 = 90. 183. From the explanation above may be derived Rule I. To find the least common multiple of two or more numbers : Take the prime factors of one of the numbers ; with these take such prime factors of each of the other numbers in succession as are not contained in any preceding number, and find the product of all these prime factors. 184. Oral Exercises. What is the least common multiple a. Of 4, 5, and 8? c. Of 6, 14, and 21 ? b. Of 6, 8, and 12 ? d. Of 3, 4, and 5 ? When several numbers are prime to each other, what must their least common multiple equal ? 185. Examples for the Slate. Find the least common multiple 63. Of 8, 18, 20, and 21. 64. Of 3, 5, 12, 36, and 45. NOTE. When one of the given numbers is contained in another, the smaller may be disregarded in the operation; thus, in the preceding example, 3, 5, and 12 may be rejected. Why? Find the least common multiple 65. Of 18, 36, 60 and 72. 68. Of 18, 32, 48, and 52. 66. Of 12, 42, 56, and 70. 69. Of 16, 28, 35, and 63. 67. Of 13, 28, 39, and 49. 70. Of the nine digits. |