88. The Greatest Common Divisor of two or more numbers is the greatest factor found in each of them. Thus, 6 is the greatest common divisor of 18 and 24. 89. Numbers are Prime to each other when they have no common factor or divisor. Thus, 9 and 14 are prime to each other. 90. Principle. The greatest common divisor of two or more numbers is the product of all their common prime factors. WRITTEN EXERCISES. 49. What is the greatest common divisor of 84 and 132 ? duct of these, 2 X 2 X 3 or 12, is the greatest common divisor required. 50. What is the greatest common divisor of 36, 81, 135 ? 51. What is the greatest common divisor of 24, 42, 54, and 60? 91. Rule for finding the Greatest Common Divisor. Separate the numbers into their prime factors, and find the product of all such as are common to the numbers. What is the greatest common divisor 60. Having three rooms, the first 12 feet wide, the second 15 feet, and the third 18 feet, I wish to purchase a roll of the widest carpeting that will exactly fit each room without any cutting as to width. How wide must it be? 92. When the numbers are large, or cannot readily be separated into factors, Of two numbers divide the larger by the smaller, and the last divisor by the last remainder, until nothing remains. The final divisor is the greatest common divisor. If more than two numbers are given, find the greatest common divisor of two of them, then of this divisor and a third number, and so on. 61. What is the greatest common divisor of 247 and 323? If 76 will exactly divide 247, it will be the greatest common divisor of 76 and 247, and therefore of 247 and 323. It will not exactly divide 247, for there is 19 remainder. If 19 will exactly divide 76, it will be the greatest common divisor of 19 and 76, and of 76 and 247, and of 247 and 323. 19 will exactly divide 76, and therefore it is the greatest common divisor of 247 and 323. Find the greatest common divisor of 62. 336 and 480. 67. 2145 and 3471. 78. Name four numbers of which 4 is a factor. 79. 15 is an exact number of times what two numbers? 80. What is the least number that will contain both 3 and 7 an exact number of times? 81. What is the least number that both 4 and 9 will exactly divide? 93. A Multiple of a number is any number divisible by it (Art. 79, note). Thus, 5, 10, 15, etc., are multiples of 5. 94. A Common Multiple of two or more numbers is any number divisible by each of them. Thus, 48 is a common multiple of 4, 8, and 12. 95. The Least Common Multiple of two or more numbers is the least number divisible by each of them. Thus, 24 is the least common multiple of 4, 8, and 12. 96. Principle. The least common multiple of two or more numbers contains each of the prime factors of those numbers, but no others. WRITTEN EXERCISES. 82. What is the least common multiple of 21, 35, and 45? greatest number of times it occurs in any number, to be 3 × 3 × 5 × 7, or 315. This is the least common multiple, because it contains each prime factor of the numbers, but no other (Art. 96). Or, the different prime factors may be found by the second process annexed, the two methods being alike in principle. 83. What is the least common multiple of 7, 14, 15, and 21? 84. What is the least common multiple of 18, 28, 30, and 42 ? 97. Rule for finding the Least Common Multiple. Separate the numbers into their prime factors. Take the product of all the different factors, using each factor the greatest number of times it occurs in any number. NOTE 1. The following rule is sometimes used. Strike out any of the given numbers that are factors of any of the others, and divide the remaining numbers by any prime factor common to two or more of them. Strike out from the resulting quotients and undivided numbers all that are factors of any of the rest, and divide as before. Thus proceed until no two of the remaining numbers have a common factor. The product of the divisors and remaining numbers will be the least common multiple required. NOTE 2. The least common multiple of numbers prime to each other is their product. Find the least common multiple 85. Of 21, 33, 56. 87. Of 66, 88, 110. 89. Of 8, 18, 24, 36. 90. Of 7, 25, 12, 41. 93. What is the least common multiple of 24, 96, 100, and 144? 94. What is the least common multiple of 4, 11, 18, 20, 36, and 48 ? 95. What is the least sum of money that can be exactly expended for sheep, cows, or oxen, at $5, $35, and $50 each, respectively? MISCELLANEOUS EXERCISES. 96. What are the prime factors of 2520 ? 97. How many times are 2 and 3 respectively factors in 5760? 98. What is the greatest factor common to 689 and 1573 ? 99. Find the sum of all the prime numbers between 70 and 100. 100. Employ cancellation in dividing 14 × 15 × 16 × 24 X 48 X 60 by 7 x 30 x 8 x 8 x 6 x 12 x 3. 101. How many times is the greatest common divisor of 48, 36, 72, 24 contained in their least common multiple ? 102. Find the sum of the composite numbers between 100 and 120, inclusive. 103. What is the difference between the greatest common divisor and the least common multiple of 160, 352, and 992 ? 104. Divide 1008 by 168, using prime factors and cancellation. 105. When hay is $ 24 a ton, how many barrels of flour, at $8, will exactly pay for 35 tons of hay? QUESTIONS. 78. What are the factors of a number? 79. What is a composite number? 80. What is a prime number ? 81. What is a prime factor? 83. How do you find the prime factors of a number? 84. What is cancellation? 85. What are principles of cancellation? 86. How is cancellation performed ? 87. What is a common divisor of two or more numbers? 88. The greatest common divisor of two or more numbers? 90. What is a principle of common divisors? 91. How is the greatest common divisor found? 93. What is a multiple of a number? 94. A common multiple of two or more numbers? 95. The least common multiple of two or more numbers? 96. A principle of multiples? 97. How is the least common multiple found? |