11.m A boy sold a merchant 4 oranges at 6 cents apiece, and 9 apples at 4 cents apiece, and took his pay in cigars at 5 cents a piece; how many cigars did he get? Ans. 12 cigars. 12. A farmer sold 27 cords of wood at $5 a cord, and 47 hogs at $7 apiece, and took in exchange flour at $8 a barrel; how many barrels did he get?

Ans. 58 barrels.

13.m How many minutes will it take 2 boys to remove 80 rails if one boy removes 6 rails, and the other 4 rails, every minute?

14. Two railway trains are 540 miles apart, and travel towards each other at the rates of 25 miles and 20 miles per hour; how many hours before they will meet? Ans. 12 hours.

15.m If a horse walks 35 miles in 7 hours, how far will he walk in 11 hours at the same rate?

16. If a railway train goes 304 miles in 16 hours, how far will it travel in 21 hours at the same rate? Ans. 399 miles.

18.

17.m How much is of the sum of 12 and 8? What is the average width of a field which is 332 yards wide at one end, and 478 yards wide at the other? Ans. 405 yards. 19.m How much is of the sum of 11, 9, and 4? A farmer killed 3 hogs; one weighed 165 pounds, another 173 pounds, and the third 181 pounds; what was the average weight of the three hogs?

Ans. 173 pounds.

21.m Henry started on a journey of 62 miles, and traveled at the rate of 5 miles a day for 4 days; how many days will it take him to complete the journey if he goes at the rate of 6 miles a day?

22. A laborer engaged to remove 4703 bricks. After

working 24 hours, removing 125 bricks each hour, how many hours will it take him to complete the work if he removes 131 bricks per hour?

Ans. 13.

23.m Frank sold 10 oranges at 6 cents apiece, and with the money bought apples at 5 cents apiece; how many apples did he get?

24. A man sold 144 horses at $135 apiece, and invested the money in land at $18 per acre; how many acres did he get? Ans. 1080 acres.

25.m A boy spent 42 cents for cakes, and 1-sixth as much for nuts; what are the c parts of what he spent in all? How much did he spend?

26. A farmer has 306 acres in one field, and 1-eighteenth as much in another field; how much has he in both fields? Ans. 323 acres.

144. QUESTIONS FOR REVIEW.

What is: 1. Division? 2. The dividend? 3. The divisor? 4. The qnotient? 5. The sign of division?

Name the three principles of division. Can division be effected by subtraction?

What is the relation of division to multiplication? What do we find by division?

What is meant by: 1. Short division? 2. Long division? Give the rule for each.

How do we find: 1. The half of a number? 2. The third of a number? 3. The fourth? 4. The fifth? etc.

DIVISORS AND MULTIPLES.

145. A Divisor of a number is one of its factors.*

Thus, 4 is a divisor of 12, since 4 × 3 = 12. Also, 1, 2, 3, 4, 6, and 12 are divisors of 12, since each is contained in 12 an exact number of times.

1o. 1 is a divisor of every number.

2°. Every number is a divisor of itself.

Any number is exactly divisible:

3°. By 2, if its last figure is divisible by 2.

4°. By 4, if its last two figures are divisible by 4. 5°. By 8, if its last three figures are divisible by 8. 6°. By 3 or 9, if the sum of its figures is divisible by 3 or 9.

8°. By 6, if it is divisible by 2 and 3.

MENTAL EXERCISES.

How many times is:

1. 1 contained in 11? 1 contained in 143?

2. 13 contained in 13? 97 contained in 97?

Which of the numbers 2, 3, 4, 5, 6, 8, 9, and 10 are divisors of: 250? 3056? 4581? 1722? 45460? 761? 23202? 45128? 37301? 371820?

The terms numbers, divisors, multiples, and factors, as here used, denote integers.

146. A Common Divisor of two or more numbers is any number that will exactly divide each of them.

Thus, 2 and 4 are common divisors of 8 and 12, since they divide each of them without a remainder.

EXERCISES.

1. What are the common divisors of 6, 12, and 24? Ans. 2, 3, and 6.

2. What are the common divisors of 8, 16, and 24? Ans. 2, 4, and 8.

3. What are the common divisors of 12, 18, and 24?

4. What are the common divisors

Ans.

Ans. 2, 3, and 6. of 36, 54, and 72? 2, 3, 6, 9, and 18. 5. What are the common divisors of 24, 36, and 48? Ans. 2, 3, 4, 6, and 12.

147. The Greatest Common Divisor of two or more numbers, denoted by G. C. D., is the greatest number that will exactly divide each of them.

Answer these five questions by referring to the preceding exercises: What is the G. C. D. of 6, 12, 24? Of 8, 16, 24? Of 12, 18, 24? Of 36, 54, 72? Of 24, 36, 48?

148. PRINCIPLE.-The G. C. D. of several numbers is the greatest factor common to all.

Thus, the c factors of 12 are 2, 2, 3,

the c factors of 18 are 2, 3, 3.

Now, from each we may take the factor 2, then the factor 3, and as no equal factors remain, 2 and 3 are the c factors of the G. C. D. of 12 and 18.

149. What is the G. C. D. of 18, 27, and 36?

OPERATION.

3)18. 27. 36.

3) 6. 9. 12.

EXPLANATION.—First divide 18, 27, and 36 by any number that will divide each of them, as 3. Then divide the quotients 6, 9, and 12, by any number that will exactly divide each of them, as 3. Now there is no number except 1 that will divide the quotients 2, 3, 4. Hence, 3 and 3, the two divisors, are the c factors of the G. C. D.; that is, the G. C. D. is 9.

WRITTEN EXERCISES.

2. 3. 4.

Ans. 6.

Ans. 6.

Ans. 7.

Ans. 18.

Ans. 24.

1. What is the G. C. D. of 12 and 30? 2. What is the G. C. D. of 24, 30, and 54? 3. What is the G. C. D. of 35, 56, and 70? 4. What is the G. C. D. of 18, 36, and 72? 5. What is the G. C. D. of 48, 72, and 144? 6. What is the G. C. D. of 32, 48, 64, and 160? Ans. 16. 7. What is the greatest common divisor of 60, 90, 150,

150. A Prime Number is a number which has no divisors except itself and 1.

Thus, 1, 2, 3, 5, 7, etc, are prime numbers.

151. A Composite Number is a number which has other divisors than itself and 1.

Thus, 4, 6, 8, 9, 10, etc., are composite numbers.