36. A man bought 4 horses at $116 apicce and 2 colts at $56 apiece, and paid for them in flour at $12 a barrel; how many barrels of flour did it require to make the pay

ment?

37. A man travels due north for 7 days at the rate of 37 miles a day; he then returns on his path at the rate of 29 miles a day; how far is he from the starting point at the end of 12 days travel?

38. A man bought 742 acres of land at $18 an acre; he sold at one time 211 acres at $22 an acre and at another time he sold 184 acres at $25 an acre; at what rate per acre must he sell the rest to gain $3,867 ?

39. A. bought a farm for $3,612; he sold half of it at $56 an acre and received for it $2,408: how many acres did he buy and what did he give per acre?

40. How many horses worth $112 apiece can be bought for 28 oxen worth $63 each, 52 cows worth $42 each, 175 sheep worth $6 each, and $2,394 in cash?

(35.) What is division? Define the dividend; the divisor; the quotient; and the remainder. When is division exact? What are factors (36.) What does the sign of division indicate when placed between two numbers? In what other ways may division be indicated? (37.) What is the relation between multiplication and division? (38.) State the leading principle of division. (39.) What is short division? What is long division? (40.) Give the rule for short division. Method of Proof? (41.) Give the rule for long division. (42.) How do you divide by a composite number? What is the method of determining the true remainder? How do you divide by 10, 100, 1,000, etc.? How do you divide by a number that ends in ciphers? How do you divide when both dividend and divi. sor terminate in ciphers? (43.) What are the fundamental opera tions of arithmetic ?

V. FACTORING AND CANCELLING.

DEFINITIONS.

44. A Factor of a number is one of its exact integral divisors, (Art. 35). Thus, 2, 3, and 4, are factors of 12. 45. A Composite number is a number composed of two or more Integral Factors besides the factor 1 (Art. 34). Thus, 15 is a composite number, because 15=3 × 5.

A Prime number is one that cannot be separated into any integral factors except 1 and the number itself. Thus, 2, 3, 5, etc., are prime numbers.

46. Factoring is the operation of separating a number into integral factors.

A Prime Factor is one that is a prime number; a Composite Factor is one that is a composite number. Composite factors can often be separated into prime factors. Thus, we have, 24 = 2 × 12 : 2 x 2 x 6 = 2 × 2 × 2 × 3.

MENTAL EXERCISES.

1. What is the product of 2 and 3? What are the factors of 6? of 9? of 15? of 77 ? of 121 ? of 144?

2. What is the continued product of 2, 3, and 7? of 2 and 3? of 2 and 7? of 3 and 7? What are the prime factors of 42? What are the composite factors of 42 ? In how many ways may 42 be factored ?

NOTE.-Because every number is the product of 1 and of the number itself, these numbers are not specially considered in the operation of factoring.

3. What are the prime factors of 4? of 9? of 27? of 81 ? of 121 ? of 143 ? of 187? of 198? of 225 ?

NOTE. The product of two or more factors that are equal is called a power. The name of the power depends on the number of

3

equal factors. Thus, 3 × 3, or 9, is the second power, or the square of 3; 3 × 3 × 3, or 27, is the third power, or the cube of 3; 3 × 3 × 3 × 3, or 81, is the fourth power of 3; and so on. Every lower power of a number is a factor of a higher power of the same number.

. 4. What are the factors of 81? of 27? of 9? How many prime factors has 81? How many has 27? How many has 9? What is the square of 6? the third power of 5? the fourth power of 4? the fifth power of 2 ?

5. What is the second power of 10? the third power of 10? the fourth power of 10? How many ciphers are required to write the square of 10? the cube of 10?

6. What is the fifth power of 10? How many ciphers in the fifth power of 10? What power of 10 is 100? 1,000? 10,000? 100,000? 1,000,000 ?

PRINCIPLES OF FACTORING.

47. The operation of resolving a number into prime factors depends on the following principles:

1°. A number is equal to the continued product of all its prime factors.

2o. If a number is divided by one of its prime factors, the quotient is equal to the continued product of all the others.

OPERATION OF FACTORING.

48. Let 210 be separated into prime factors.

EXPLANATION.-We first divide by 2, which is a prime factor; we next divide the first quotient by 3, which is also a prime factor; we then divide the second quotient by 5, and find 7 for a quotient. The numbers 2, 3, 5, and 7 are the required factors; that i

OPERATION.

2)210

3)105

5)35

In like manner other composite numbers may be factored; hence, the following

RULE.

Divide the given number by one of its prime factors; then divide the quotient by one of its prime factors; and so on, till a quotient is found that is a prime number; the several divisors and the last quotient are the required factors.

NOTE.-It will be found convenient to begin the division with the smallest prime factor.

EXAMPLES.

Resolve the following numbers into their prime factors: 1. 42. Ans. 2 x 3 x 7.

NOTE.-If there are more than two factors in any indicated product, the sign of multiplication may be replaced by a simple dot: thus, 2. 3. 7 is equivalent to 2 × 3 × 7.

2. 180. Ans. 2.2.3.3.5. 5. 770. Ans. 2.5.7.11. 3. 378. Ans. 2.3.3.3.7. 6. 1,575. Ans. 3.3.5.5.1. 4. 330. Ans. 2.3.5.11. 7. 3,850. Ans. 2.5.5.7.11. The operation of factoring is principally performed by inspection and trial. It may sometimes be facilitated by using the following

TABLE OF PRIME NUMBERS FROM I TO 150.
7 11 13 17 19

1 2 3

5

NOTE.-If the final digit of a number is 0, 2, 4, 6, or 8, the num ber is divisible by 2.

If the sum of the digits of a number is divisible by 3, the num ber itself is divisible by 3.

If the final digit of a number is 5, the number is divisible by 5.

49. Cancellation is the operation of striking out one or more factors that are common both to the dividend and the divisor of an indicated division.

The operation is performed by drawing a line across the factor that is to be struck out, or cancelled. Thus, in the 2.3.4 expression 2.3.7

the factors 2 and 3 are cancelled.

OBJECT AND PRINCIPLES OF CANCELLATION.

50. The operation of division may sometimes be shortened by cancelling factors common to both dividend and This method depends on the following prin

divisor. ciples:

1o. Striking out a factor of a number is equivalent to dividing the number by that factor.

2°. If both dividend and divisor are divided by the same number the quotient is not changed.

1. What is the quotient of 3 x 8 by 3 x 4? of 8 by 4? What is the effect of cancelling 3 in both dividend and divisor?