50. Cattle. The greatest beef-producing states are Texas, Iowa, and Kansas. The greatest stock yards and packing houses in the world are in Chicago.

1. If Texas has 4,533,897 head of cattle, valued at $17 a head, how much are they all worth?

2. If Kansas has 2,076,489 head of cattle, valued at $27 a head, how much are they all worth?

3. If Iowa has 2,163,584 head of cattle, valued at $31 a head, how much are they all worth?

4. If this country sold 55,553,750 lb. of canned beef in a certain year, at 9 ct. a pound, what was the total value? 5. In a year when the census showed 69,335,832 cattle in this country, their value was estimated at $1,525,388,304. What was the average value per head?

6. In a year when the census showed 18,108,666 cows kept for milk, their value was estimated at $543,259,980. What was the average value per head?

7. The number of cattle being raised for food in the United States in 1880 was 21,231,000, and our population was 50,155,783. In 1900 the number of cattle was 27,610,054, and our population was 76,303,387. What was the increase in cattle and in population in the 20 years?

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, how much does it cost a pound delivered in New York?

FACTORS AND MULTIPLES

ORAL EXERCISE

1. What do you mean by the factors of a number?
2. Name two factors of 35; of 77; of 26; of 34.
3. Name three factors of 30; of 70; of 66; of 110.
4. Name a factor that is common to both 35 and 45.
5. Name two factors common to both 30 and 48.

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, 3 ?

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. or 6, is the greatest common divisor.

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 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. 3. Find the greatest common divisor of 96, 60, and 36.

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? it a factor of any number

of any number of 10's? Then is

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, 206,340, 475,633, 2,124,816, 4,216,872, 15,681,123?