yards on Thursday, and sold 1940 yards of it. How many yards remained?

27. A man bought a farm, paying $1690 down. After making three other payments of $945, $765 and $2740, the amount unpaid was $3628. He sold the farm for $8278. What was his loss?

28. A merchant's store was insured for $21650, and his goods for $17645. The store was valued at $25000, and the goods at $22564. Both were destroyed by fire. What was his loss?

29. A, B, C and D formed a partnership. A invested $720 more than B; C invested $1280 less than B, and D invested $945 less than A and C together. What was the total investment if B invested $2450?

30. A speculator bought 200 acres of land at $45 per acre, and afterwards sold 150 acres of it for $11550; the balance he sold at a gain of $5 per acre, and received in payment $250 cash, and the balance in sheep at $5 each. How many sheep did he receive? What was his profit on the land?

FACTORING

60. An Exact Divisor of a number is a number that will contain it an integral number of times.

61. A Factor of a number is an integral divisor of the number. The factors of 18 are 2, 3 and 3; of 30 are 2, 3 and 5.

62. A Prime Number is one which cannot be divided or separated into factors, except 1 and that number; as 3, 5, 7, 11 and 13.

63. A Composite Number is one which can be resolved or separated into integral factors; as 4, 6, 9, 10 and 12.

64. An Even Number is one that is exactly divisible by 2; as 8, 16, 24 and 30.

65. An Odd Number is one that is not exactly divisible by 2; as 5, 9, 7, 15 and 21.

66. A Prime Factor is one that is a prime number; as 3, 11, 17 and 23.

67. Factoring is finding the factors of a number.

1. What are the prime factors of 42?

From the preceding definitions and example we have the following:

To Separate a Number Into Its Prime Factors

a. Divide the given number by any prime factor.

b. Divide the quotient, if a composite number, by any prime factor and so continue dividing until the quotient is a prime number.

c. The divisors and the last quotient are the prime factors.

68. Since dividing both dividend and divisor by the same number does not change the quotient, we may strike out or reject equal factors from both dividend and divisor without affecting the result.

69. Cancellation is striking out equal factors from both dividend and divisor. Cancellation is a short method of division. 70. The Sign of cancellation is an oblique line, /; when drawn across a number, it shows the number has been cancelled; as, £, $, $ and $.

The most common method is to write the dividend above a horizontal line and the divisor beneath it, although a vertical line with the divisor on the left and the dividend on the right is sometimes used.

If the dividend or divisor is a composite number it should first be resolved into its prime factors, before beginning the operation of cancellation.

1. Divide 4 X 8 × 3 × 5 × 7 by 2 × 3 × 4 × 5.

We therefore have the following:

To Find a Quotient by Cancellation

a. Write the numbers composing the dividend above a horizontal line and those composing the divisor below it.

b. Cancel all factors common to both dividend and divisor.

c. Multiply the remaining factors of the upper numbers or dividend together, and the remaining factors, if any, of the lower numbers or divisor together. Divide the upper product by the lower and the result will be the quotient desired.

NOTES.-1. Rejecting a factor from any number is dividing the number by that factor.

2. When a factor is cancelled, the unit, 1, is supposed to take its place.

3. One factor in the dividend will cancel only one equal factor in the divisor.

4. If all the factors or numbers of the divisor are cancelled, the product of the remaining factors of the dividend will be the quotient.

5. By many it is thought more convenient to write the factors of the dividend on the right of a vertical line, and the factors of the division on the left.

Solve the following problems by cancellation:

2. Divide 3 X 7 × 4 × 5 × 11 by 5 × 7 × 4 × 3.
8. Divide 2 X 4 × 6 × 8 × 10 by 2 × 4 × 5 × 6.
4. Divide 8 × 6 × 12 × 15 by 5 × 3 × 4 × 2 × 3.
5. Divide 9 X 8 X 6 X 18 by 2 X 3 X 4 X 3 X 2.
6. Divide X6 X 3 X 14 by 2 X3 X 2 X 7.

7. Divide 4 X 6 X 12 X 15 X 8 by 2 × 3 × 6 × 5 × 4.
8. Divide 3 × 6 × 9 × 12 × 24 by 3 × 4 × 6 × 8 × 5.
9. How many yards of calico at $.08 per yard will pay for 14

bushels of turnips at $.28 per bushel?

10. If 8 yards of cloth cost $32, what will 19 yards cost? 11. How many hogs at $9 each will pay for 36 barrels of flour at $6 each?

12. I exchanged 48 dozen of eggs at 10 cents per dozen for sugar at 8 cents per pound. How many pounds did I get?

13. I bought 8 boxes of boots, each containing 9 dozen pairs at $18 per dozen, and gave in payment 12 loads of wheat of 54 bushels each. What did I receive per bushel for the wheat?

14. A cubic yard is 3 feet deep, 3 feet long and 3 feet wide. How many cubic yards in 24 feet long, 18 feet wide and 6 feet deep?

GREATEST COMMON DIVISOR

71. A Common Divisor of two or more numbers is a number that will divide them integrally.

72. The Greatest Common Divisor of two or more numbers is the greatest number that will divide them integrally, or is the product of all their common prime factors.

Numbers are said to be prime to each other when they have no common divisor.

If a number divides two or more others, it will also divide their sum and difference, and also the sum and difference of any product of them, because it divides them when they are taken separately. Hence if one number divide the whole of another number, and also one part of it, it will divide the other part also. Thus, 9 divides 45 and 27, and also their difference, 18.

1. Find the Greatest Common Divisor of 84 and 203.

SOLUTION

84) 203 (2 168

35)84(2
70
14)35(2
28

EXPLANATION.-Write the smaller number as the divisor and the larger as the dividend. We know that the G. C. D., whatever it is, must divide 84, and if it divides 84 it will also Idivide twice that number or 168. If it divides 203 and 168 it must divide their difference, 35, according to the principle laid down above. Repeating this argument until the end, it must divide 7 and 14, and as 7 is contained in 14 7)14(2 without a remainder, 7 is the largest number that will divide itself and 14, and hence is the G. C. D.

14

From the preceding principles and explanations we have the following rule:

To Find the Greatest Common Divisor

a. Divide the greater number by the less, and then the less by the remainder, until nothing is left.

b. The last divisor will be the Greatest Common Divisor.