51. The stock yards. The Chicago stock yards are the center of the great beef industry. From the packing houses the meat is shipped in refrigerator cars, and goes to various parts of the world. WRITTEN EXERCISE 1. In one year the receipts from the beef industry of the country were $800,000,000, of which was profit. How much was it worth at $8.00 a hundred, adding $1.50 for preparing the meat for shipment? 3. If 4,489,500 cattle were shipped to some stock yards in one year, what was the average per day? 4. An eastern dealer buys 10,000 lb. of beef a month, at 14 ct. a pound, and sells it on an average of 19 ct. a pound. How much are his profits a month? 5. In 50 years the value of the beef produced annually in this country rose from $11,981,642 to $785,562,442. How much was the average annual increase in value? 6. If 700 lb. of meat are sold by a packing house for $63, and it costs $1 a hundred to ship it to New York, FACTORS AND MULTIPLES ORAL EXERCISE 1. What do you mean by the factors of a number? 6. In reducing to we found in primary arithmetic that we could divide both terms by 2, or cancel the 2. What factor do we cancel in reducing § to its lowest terms, & ? 7. Name the factors to be canceled in reducing to lowest terms,,, §, 12, 18, 18. What need have we found for factors in arithmetic? 52. Common divisor. A factor that is common to two or more numbers is called a common divisor. For example, 5 is a common divisor of 35 and 45, as we found in Ex. 4. It might be called a common factor, but common divisor is the name usually given it. 53. Greatest common divisor. The greatest factor common to two or more numbers is called their greatest common divisor (g.c.d.). For example, although 3 is a common divisor of 12, 18, and 48, the number 6 is the greatest common divisor. 54. Finding the g.c.d. The greatest common divisor of such numbers as we shall meet in fractions is easily found. For example, consider 12 and 30. We see that Therefore 2 × 3, and that 2 and 3 are the only common factors. ORAL EXERCISE 1. Name a common divisor of 20 and 22; 16 and 36. 2. Name the greatest common divisor of 75 and 100. 3. Which of these numbers have no factors excepting one and themselves: 4, 5, 6, 7, 10, 11, 12, 13? 4. In reducing 12 to lowest terms is it sufficient to cancel the common divisor 2? What kind of common divisor must be canceled in reducing a fraction to lowest terms? 55. Prime number. A number that has no factors except itself and 1 is called a prime number. For example, 7, 11, 17 are prime numbers. 56. Composite number. A number not prime is called composite. 57. Even numbers. A number which contains the factor 2 is called an even number. 58. Odd numbers. Numbers that are not even are called odd numbers. 59. Prime factors. Factors that are prime numbers are called prime factors. For example, the prime factors of 68 are 2, 2, and 17. 60. Mutually prime numbers. Numbers that have no common factors are said to be prime to each other. For example, 12 and 35 are prime to each other. WRITTEN EXERCISE 1. Make a list of prime numbers to 100; of odd numbers. 2. Find the prime factors of 54; of 48; of 64; of 77. ORAL EXERCISE 1. Is 2 a factor of 14? of 27? of 50? of 45? of 600? How do you tell whether 2 is a factor of a number? 2. Is 2 a factor of 10? of any number of 10's? Then is it a factor of any number if it is a factor of the units? 3. Is 5 a factor of 20? of 78? of 45? of 72? of 800? How do you tell whether 5 is a factor of a number? 4. Is 5 a factor of 10? of any number of 10's? Then is it a factor of any number if it is a factor of the units? 61. Divisible numbers. When we speak of one number being divisible by another we mean exactly divisible. 62. Divisibility by 2. A number is divisible by 2 if the units are so divisible. For example, 64, or 60 + 4, must be divisible by 2 if 4 is so divisible, because 60 is divisible by 2. 63. Divisibility by 5. A number is divisible by 5 if it ends in O or 5. 64. Divisibility by 3. A number is divisible by 3 if the sum of its digits is so divisible. For example, 411 is divisible by 3 because 4 + 1 +、1 is. WRITTEN EXERCISE 1. Which of these are divisible by 2: 660, 4907, 6255, 3027, 1356, 5790, 2371, 4196, 2005, 37,268, 125,474? 2. Which of these are divisible by 5: 660, 7620, 4867, 7075, 3200, 4035, 9636, 8124, 3672, 12,475, 374,465? 3. Which of these are divisible by 3: 660, 1236, 5778, 9102, 8328, 3444, 1239, 9876, 4004, 7117, 31,476, 307,983, ORAL EXERCISE 1. State the prime factors of 15, 18, 27, 35, 42, 75. 2. State one factor of 395, 123, 777, 692, 1275, 1263. 3. Which of these are divisible by 3: 77, 609, 1203? 4. Which of these are divisible by 2: 68, 4973, 2870? 65. Other tests. There are various tests of divisibility besides those on page 32. They are easily illustrated or explained, and may be given or not as the teacher prefers. The more important are the following. 66. Divisibility by 4. A number is divisible by 4 if the number represented by the two right-hand figures is so divisible. 67. Divisibility by 6. A number is divisible by 6 if it is even and if the sum of its digits is divisible by 3. 68. Divisibility by 8. A number is divisible by 8 if the number represented by the three right-hand figures is so divisible. 69. Divisibility by 9. A number is divisible by 9 if the sum of its digits is so divisible. 70. Divisibility by 11. A number is divisible by 11 if the difference between the sums of the digits in its even and odd places is so divisible. For example, 430,507 is divisible by 11, for 7 + 5 + 3 = 15, and 0 + 0 + 4 = 4, and 15 4 11. = 71. Finding prime factors. Find the prime factors of 2310. By § 62, 2 is a factor. By § 63, 5 is a factor of the other factor, 1155. By § 64, 3 is a factor of the other factor, 231. It is easy to see that 7 and 11 are factors 2)2310 5)1155 3)231 7)77 11 |