29. Three pieces of lead, weighing respectively 24 oz., 32 oz., and 40 oz., were cast without waste into balls of equal weight and as heavy as possible. What was the weight of each ball? 30. What is the length of the longest board that can be used, without waste, to build a fence around a lot which is 60 ft. long and 48 ft. wide? MULTIPLES 65. A number that a given number will exactly divide is called a multiple of the given number. Thus, 6 is a multiple of 2, since 2 will exactly divide 6. 66. A number that each of two or more numbers will exactly divide is called a common multiple of them. Thus, 12 is a common multiple of 2 and 3, since 2 and 3 will each exactly divide 12. 67. The least number that each of two or more numbers will exactly divide is called their least common multiple (1. c. m.). Thus, 6 is the least common multiple of 2 and 3, since 6 is the least number that 2 and 3 will exactly divide. Exercise 24 1. Name a multiple of 2; of 3; of 5; of 7. 2. Name three multiples of 2. 3. Name three multiples of 3. 4. Name two multiples of 7. 5. Name a common multiple of 3 and 4. Name the least common multiple of: Find the least common multiple of 60, 90, and 210. prime factor 7, in order to contain 210. Therefore, 2 × 2 × 3 × 5 × 3 × 7, or 1260 = 1. c. m. of 60, 90, and 210. The least common multiple of two or more numbers is the product of all their different prime factors each taken the greatest number of times it occurs in any one of them. 16. What is the least sum of money that can be put up in packages of either $20 bills or $50 bills? 17. What is the least weight of hay that can be put up in either 100 lb. bales or 125 lb. bales? 18. What is the least debt that can be paid in either $2 bills, $5 bills, or $10 bills? 19. What is the least number of children that can be formed in groups of 8, 12, or 20, and have none remaining? CANCELLATION 68. Cancellation is the process of crossing out a number and writing the result obtained by multiplying or dividing it. Important Principles 69. Multiplying one number and dividing another by the same number does not change their product. Thus, 6 × 10 = 60, and (6 × 2) × (10÷2), or 12 × 5 = 60. 70. Multiplying both dividend and divisor by the same number does not change the quotient. Thus, 12 ÷ 3 = 4, and (5 × 12) ÷ (5 × 3), or 60 ÷ 15 = 4. 71. Dividing both dividend and divisor by the same number does not change the quotient. Thus, 18÷6=3, and (18÷3) ÷ (6÷3), or 6 ÷ 2 = 3. 2 Thus, 5 6 × 9 × 23 10 × 3 = 45 6 × 9 × 25 read 6 times 9 times We cancel the 3 in the divisor, and the 3 from the 6 in the dividend, writing the quotient 2. We next cancel the 2 in the dividend and the 2 from the 10 in the divisor, writing the quotient 5. Then we cancel 5 from the divisor and 5 from the 25 in the dividend, writing the quotient 5. There then remain the factors 9 and 5 in the dividend; their product, 45, is the value of the expression. |