7. A confectioner put 2151 pounds of candy into boxes holding 1 pound, 3 pounds, and 5 pounds, respectively, using the same number of boxes of each kind. How many boxes were used for all the candy?

8. What number must be added to the sum of 342.807, 231.96, and 324.7 to equal the difference between 2107.62 and and 1009.006 ?

9. A certain number was divided, and 20.45 was both quotient and divisor. What was the number divided?

10. Make and solve a problem that requires the product of two numbers to be subtracted from the product of two other numbers.

11. Make and solve a problem that requires the product of two numbers to be added to the product of two other numbers and the sum divided by a certain number.

12. Divide by 37 the result obtained by adding 111 to the product of 148 and 6090.

13. A merchant bought 345 pounds of wool of one man, 3067 pounds of another, 468 pounds of another, and 384 pounds of another, and sold of it at 27 cents a pound. What did he receive for the part sold?

14. Make and solve a problem that may be indicated thus: $10-($.35+$2.20 +$6.19+ $.18).

TESTS OF DIVISIBILITY

72. The figures used in Arabic notation are called digits. Name the digits.

73. A number that can be exactly divided by 2 is an even number; e.g. 2, 4, 18.

74. A number that cannot be exactly divided by 2 is an odd number; e.g. 3, 7, 19.

75. A number is exactly divisible

a. By 2, if the digit in units' place is 0 or even; e.g. 70, since the units' digit is 0; 35,976, since the units' digit is

even.

b. By 4, if the digits in units' and tens' places are 0's; or if the number expressed by them is divisible by 4; e.g. 3100, 3976. How do you know? know? How can we tell without actual trial that 2398 is not divisible by 4?

c. By 8, if the digits in units', tens', and hundreds' places are 0's, or if the number expressed by them is divisible by 8; e.g. 11,000 and 37,112. How do you know?

76,518?

Why not

d. By 3, if the sum of its digits is divisible by 3; e.g. 24,762, since 2 + 4 + 7 + 6 + 2, or 21, is divisible by 3.

e. By 9, if the sum of its digits is divisible by 9; e.g. 397,647, since 3+9+7 + 6 + 4 +7, or 36, is divisible by 9.

f. By 5, if the units' digit is 0 or 5; e.g. 80; 115.

g. By 25, if the units' and tens' figures are O's, or if the number expressed by them is divisible by 25; e.g. 1900; 8375.

h. By 125, if the units', tens', and hundreds' figures are O's, or if the number expressed by them is divisible by 125 ; e.g. 13,000 ; 71,750.

i. By 10 or a power of 10, if it contains as many O's at the right of its significant figures as there are O's at the right of the 1 in the divisor; e.g. 390 is divisible by 10; 390,000 is divisible by 10,000.

j. By 6, if the number is even and the sum of its digits is divisible by 3; e.g. 21,108.

76. Oral

1. Test each of the following numbers for divisibility by 2, 4, 8, 3, 6, 9, 5, 25, and 125:

2. An even number will not exactly divide an odd number. Why?

3. What numbers can be exactly divided by 6? Can you tell what numbers may be exactly divided by 18?

4. If 10 will divide a given number, what other numbers will divide the same number?

5. If 8 will divide a given number, what other numbers will divide it?

6. If 125 will divide a given number, what other numbers will divide it?

IDEAS OF PROPORTION

77. Oral

1. 36 is how many times 12? 12 is what part of 36? If 12 oranges cost $.35, 36 oranges will cost how many times $.35? How much will they cost? How many oranges can be bought for $.70?

2. 125 is what part of 500? If 500 sheets of paper cost 90 cents, 100 sheets will cost what part of 90 cents? 100 sheets will cost how much? At the same rate, how many sheets can be bought for 9 cents? For 45 cents?

3. A 3-pound basket of grapes cost 10 cents. rate, what must be paid for a 12-pound basket? pounds can be bought for 50 cents?

At the same

How many

At the same

4. An automobile travels 67 miles in 4 hours. rate, how far will it travel in 8 hours? In what time will it travel 33 miles?

5. A Vermont farmer made 7 pounds of maple sugar from 23 gallons of sap. At that rate, how many gallons of sap were required for 35 pounds of sugar? How many pounds of sugar could be made from 92 gallons of sap?

6. A man is paid for his work at the rate of $17 for 44 hours' work. What does he receive for 11 hours' work? How long must he work to earn $81?

7. A Kansas farmer raised 518 bushels of wheat on 14 acres of land. That was an average of how many bushels on two acres? 259 bushels of this crop were raised on how many acres? How many acres would be required to produce 1036 bushels, at the same rate? How many bushels could be raised on 42 acres at the same rate?

8. It required $110 a week to buy food for 40 boarders at a certain boarding house. What would be the weekly cost of food for 160 boarders, at the same rate? How many persons could be fed for $11 per week?

9. If 300 quarts of milk cost $21, what will 300 gallons cost, at the same price per quart?

10. How many books at 32 each will cost as much as 405 books at 96 each?

11. In how many minutes will a steamer, going 100 rods a minute, go as far as a man will row in 28 minutes, if he rows 25 rods a minute?

FACTORS AND MULTIPLES

78. A number that exactly contains another number is a multiple of that number; e.g. 21 is a multiple of 7. It is also a multiple of 3.

79. A factor that is an integer is called an integral factor; e.g. 8 is an integral factor of 56.

80. A number that is not the product of integral factors other than itself and 1 is a prime number; e.g. 2, 3, 5, 7, 11, and 13.

81. A number that is the product of integral factors other than itself and 1 is a composite number; e.g. 16, 24, 35, 1000.

82. A factor that is a prime number is a prime factor; e.g. 13 is a prime factor of 26.

NOTE. — In finding the factors of a number it is customary to consider only integral factors.

83. Oral

1. Give the factors of 51; 45; 99; 87; 96; 69; 84; 91. 2. Name three factors of 80.

3. Name as many factors of 24 as you can.

4. Of what numbers are 3, 5, and 11 the prime factors?

5. Name four multiples of 8.

6. 132 is the product of 11 and what other factor?

7. Name all the prime numbers smaller than 132.

8. 98 is the product of three factors. Two of them are 2 and 7. What is the other?

9. Of what number are 2, 3, 5, and 7 the prime factors?

10. Give the prime factors of 35; 45; 81; 63; 57; 38; 48; 51; 108; 231; 121; 144.