36. If the sum paid for the steerage passage of 240 Germans sailing from Bremen to New York was $9708, what was the price of passage per person?

37. One season this country received German toys worth $4,500,000. If all the toys exported by Germany were worth 3.4 times this amount, find their total value.

38. How much do girls who work in a German doll factory earn in 6 days of 10 hours each, if they receive 3.5¢ per hour?

39. Our sales to the people of Great Britain one year amounted to $580,322,098, and our purchases from them amounted to $191,219,295. How much money was required to balance the account and to which country was it paid?

40. During a recent tourist season in Switzerland, 26,569 Germans stopped at Geneva hotels, 9618 Englishmen, 35,114 Swiss, 68,513 Frenchmen, 14,177 Americans, and 23,094 others. How many visitors were there?

41. The value of 15,000,000 pounds of chocolate exported from Switzerland in a year was $5,100,000. The value of that exported to Great Britain was $2,125,000; of that exported to the United States, $578,000. How many pounds were exported to each of these countries?

42. One year 388,500,000 bushels of grain were sent from Russia. Find its value at 64 per bushel.

43. India obtained in a season 222,200,000 pounds of tea from the 524,500 acres under tea cultivation. Find to the nearest tenth of a pound the average yield per acre.

44. On a South African ostrich farm of 48 acres there were 5 ostriches per acre. If each bird produced $28.75 in feathers

during the year, what was the income?

45. Find the total length of railways in the world, divided among the continents thus: America, 270,386 miles; Europe, 187,776; Asia, 46,592; Africa, 15,649; Australasia, 16,702.

FACTORS AND DIVISORS

101. 1. How many are 3 × 4? 6 × 2? 3 × 2 × 2

2. Name two numbers whose product is 16; 24; four numbers whose product is 16; 24.

102. The integers that multiplied together produce a given number are called its factors.

103. An integer that will divide a number without having a remainder is called an exact divisor of the number.

The factors of a number are exact divisors of it.

2, 3, 4, and 6 are exact divisors of 12. Divisible means exactly divisible.

104. Two or more numbers that are divisible by the same number are said to have a common divisor, or a common factor.

105. A number that has no factors except itself and 1 is called a prime number.

5, 7, 11, are prime numbers.

106. A number that has other factors than itself and 1 is called a composite number.

4, 9, 12, are composite numbers.

107. 1. Tell which of the following are divisible by 2: 2, 4, 5, 8, 10, 13, 17, 18, 20, 21, 42, 50.

2. Which of these numbers are not divisible by 2?

108. A number that is divisible by 2 is called an even number; a number that is not divisible by 2 is called an odd number.

O is regarded as an even number, 1 as an odd number.

DIVISIBILITY OF NUMBERS

109. The figures that are used to represent a number are called its digits.

The digits of 358 are 3, 5, and 8.

110. Let the student illustrate with numbers each of these useful tests of divisibility.

A number is divisible by

2, if the units' figure is 2, 4, 6, 8, or 0.

5, if the units' figure is 5 or 0.

3, if the sum of the digits is divisible by 3.

9, if the sum of the digits is divisible by 9.

EXERCISES

111. By applying the preceding tests tell which of the numbers 2, 5, 3, 9, are exact divisors of :

1. By 6, if it is even and the sum of its digits is divisible by 3.

2. By 4, if its two right-hand digits are 0's or if the number expressed by them is divisible by 4.

3. By 25, if its two right-hand digits are 0's or if the number expressed by them is divisible by 25.

4. By 8, if its three right-hand digits are 0's or if the number expressed by them is divisible by 8.

5. By 11, if the difference between the sums of its alternate digits is zero or is divisible by 11.

EXERCISES

113. Find which of the numbers 2, 3, 4, 5, 6, 8, 9, 11, 25,

Illustrate the following with numbers:

13. If an even number is divisible by an odd number, it is divisible by twice that number.

14. An exact divisor of a number is an exact divisor of any number of times that number.

15. An exact divisor of each of two numbers is an exact divisor of their sum and of their difference.

FACTORING

114. 1. What numbers are exact divisors of 18?

If 3 is taken as one of two factors of 18, what is the other factor? How is it found?

2. One of two factors of 42 is 3. What is the other factor? In separating a number into two factors, any exact divisor may be taken for one factor and the quotient for the other.

115. The process of separating a number into its factors is called factoring.

116. Factors that are prime numbers are called prime factors. 2, 2, and 3 are the prime factors of 12.

117. When numbers have no common factor except 1 they are prime to each other.

4 and 21 are prime to each other, though neither of them is a prime number.

118. A small figure written at the right of a number and a little above, to indicate how many times the number occurs as a factor, is called an exponent.

In 16 = 24, the 4 is an exponent, indicating that 2 occurs 4 times as a factor of 16; that is, 24 means 2 × 2 × 2 × 2.

Hence the factors of 2295 found

are 5, 3, 3, 3, and 17; that is, 2295

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=

are 5, 9, 3, and 17, but the prime factors 5 x 38 x 17.

2. Find the prime factors of 7000; of 2880; of 8250.

3. Factor all the composite numbers from 1 to 100 and make