52. What is the dividend?

53. What is the divisor?

54. What is the quotient? To what term in division does the minuend correspond? To what does the subtrahend correspond? How many times is 6 contained in 18? How many times can 6 be subtracted from 18? Has the quotient 3 any corresponding number in subtraction? In subtraction, how many times is the subtrahend taken from the minuend? In division, how many times is the divisor contained in the dividend? To what does the remainder in subtraction correspond in division?

55. What is the remainder in division? What is the remainder in subtraction? 56. What is the sign of division? Illustrate the use of the sign of division. Is the sign placed before, or after, the divisor? What sign is placed between the divisor and quotient, when it is to be shown that 18 divided by 6 is 3?

57. In what does the fractional sign of division consist? How are the numbers written? Which is the dividend? Which is the divisor? Illustrate the use of both signs, and show their difference.

58 & 59. What is the difference between short division and long division? 60. Give and illustrate the axiom of division.

61. Give an example in which the dividend is abstract; one in which it is concrete When the divisor is abstract, what is the quotient like? When the divisor is concrete, what is the quotient?

62. Give the analysis of a problem in short division. Repeat the rule. State the method of proof. What is the remainder in division? How can a remainder be changed to a fractional quotient? What is the answer called in division? What is to be done with the fractional quotient?

63. How may a number be divided by 10, 100, 1,000, &c. By this means change mills to dollars, cents to dollars, mills to cents.

64. What is the method of long division? Repeat the rule for long division. What is the first step in long division? The second? The third? The fourth? When the next figure is brought down to the remainder, and it will not then contain the divisor, what is to be done? If the product of the divisor and the quotient figure is more than the partial dividend, what does it show? If any remainder equals or exceeds the divisor, what does it show? What is the opposite of division? How can division be proved? How can multiplication be proved? How does multiplying the divisor affect the quotient? How does multiplying the dividend affect the quotient? How does dividing the divisor affect the quotient?

65. What is the currency of the United States? When was it adopted? ANS.-1786. What are the denominations? In business, how are eagles read? How are dimes read? Which denomination of United States money is the unit? How many cents make a dollar? How many mills make a cent? A dime? A dollar?

66. How many cents in a half-dollar? In a quarter-dollar? In three quarterdollars? How many cents in 1 eighth of a dollar? In 1 tenth? In 1 fifth? In 2 fifths? A mill is what part of a cent? A cent is what part of a dollar? Into how many parts is a cent divided? What are the parts called?

67. What are coins? Of what are coins made? What is the place called where they are made? What metals are called precious? Which precious metal is worth the most? What are the gold coins of the United States? The silver coins? The nickel? The bronze?

68. What is meant by paper money?

69. What is a bill or invoice of goods? By whom are bills given? What is their use? What is a debtor? What is a creditor? When is a bill called an account? Write a bill on the blackboard. Write an account. What is it to receipt a bill? Is the bill receipted by the debtor, or by the creditor? How is a bill receipted by a clerk or agent?

GENERAL QUESTIONS IN UNITED STATES CURRENCY. How are dollars reduced to cents?

Which of the fundamental rules is used to reduce dollars to cents? Cents to mills? Mills to cents? Mills to doilars? Cents to dollars? Cents to mills? Give the formula for reducing $2 to cents. 500 cents to dollars. 4 cents to mills. 80 mills to cents. 5,000 mills to dollars. Recite the rule for reducing dollars to cents. Dollars to mills. Mills to cents. Mills to dollars. Cents to dollars. Cents to mills.

How are numbers in Federal money added? Can dollars be added to cents? Cents to mills? Mills to dollars? How are the numbers written? Why? What is the answer called? How may the answer be proved? How many places do mills take? Cents? Where is the decimal point placed in the answer? sign is placed before dollars?

What

What is subtraction of United States money? How are the numbers written? Which is the subtrahend? Which the minuend? Which the remainder? Can cents be taken from dollars? Mills from cents. Cents from cents? Show at the blackboard how 1,000 mills can be taken from a thousand cents. A thousand mills from a thousand dollars. A thousand cents from a thousand dollars. How may subtraction be proved? What is the opposite of subtraction?

What is multiplication? What do we call the number to be multiplied? Show at the blackboard how 1 dollar 1 cent and 1 mill can be multiplied by 100? Which is the multiplicand? Which the multiplier? Which the product? How should the product be written? The cost of 2 apples is how many times as much as the cost of 1 apple? 20 cows are how many times as many as 1 cow? The cost of 20 cows is how many times as much as the cost of 1 cow? How may multiplication be proved?

What is division? What do we call the number to be divided? What do we call the number by which we divide? What is the answer called? When the dividend is divided into equal parts, of what denomination is the quotient? Is it like, or unlike, the dividend? When the dividend is cents, what is the quotient? When the dividend is mills what is the quotient? When we want to find how many times one sum of money is contained in another, what must first be done? ANS.-If they are unlike, reduce them to the same denomination. If the divisor is mills, and the dividend is cents, what must be done? What if the divisor is dollars and cents, and the dividend is dollars? If the divisor is cents and mills, and the dividend is dollars and cents? If the divisor is mills, and the dividend is dollars? If the dividend is cents, and the divisor is dollars? How many times are 6 cents contained in 6 dollars? 20 mills in 20 dollars? 5 dimes in 5 eagles? 1 mill in 1 eagle? 1 dime in 1 dollar? Divide 20 dollars into 5 equal parts? Into 4 equal parts? Into 3 equal parts? ANS.-$6.666+. Why reduce the dollars to mills before dividing? ANS. - That the remainder may be mills, and therefore of small value. What does the sign plus after the answer show? What is 1 third of 30 cents? 1 third of 10 mills? What is it customary in business to do with the remainder of mills? ANS. .- When it equals or exceeds 5, add ONE to the mills; if less, reject it as of no value. In the final answer, what is done with the mills? ANS.When the mills equal or exceed 5, add ONE to the cents; if less, reject them.

SECTION VII.

THE GREATEST COMMON DIVISOR.

NOTE TO THE TEACHER. —There are many who prefer to teach compound numbers, and their application to the transactions of the shop, store, and market, before common and decimal fractions. This book is so arranged, that no difficulty will be experienced by the teacher or pupil in omitting the latter subjects (Sections VII., VIII., IX.), and in taking them up after denominate numbers.

THE practical value of the greatest common divisor and the least common multiple is very slight indeed, as they are seldom used, except in school-books. Cancellation is of more utility; but in business transactions its importance has been overrated.

When 2 and 3 are multiplied together, the product, 6, is said to be a composite number; and the numbers multiplied together are said to be factors of 6. A number that cannot be so produced by the multiplication of factors is said to be a prime number, and can have no divisor; as, 2, 3, 7, 11, &c. Since 2 and 3, the factors of 6, are prime numbers, they are said to be the prime factors of 6.

70. A Composite Number is one produced by the multiplication of two or more factors.*

71. A Prime Number is one that cannot be produced by the multiplication of factors. Numbers are said to be prime to each other when they have no common factor or divisor.

72. A Factor, or Divisor, of a number is one. of the numbers which multiplied together will produce it. When the factor is a prime number, it is said to be a prime factor.

only.

The words factor and divisor refer to factors and divisors that are integral

73. TO RESOLVE A NUMBER INTO ITS PRIME FAC

TORS.

PROBLEM.

PROCESS.
260

2 30

3 15

5

602 X2 X3 X5

What are the prime factors of 60?

ANALYSIS. - (1.) By trial, 60 is found to be the product of the prime factor 2, and the composite factor 30.

(2.) 30 is the product of the prime factor 2, and the composite factor 15.

(3.) 15 is the product of the prime factors 3 and 5.

factors of 60 are found to be 2 × 2 × 3 × 5.

RULE. - Divide by the least prime factor, and each quotient by its least prime factor, until a quotient is found which is a prime factor. The several divisors and the last quotient are the factors required.

What are the prime factors of the following numbers?

The numbers 8 and 12 can both be divided by 2 and 4: hence 2 and 4 are said to be common factors, or common divisors, of 8 and 12. Since 4 is the greatest number that will divide both 8 and 12, 4 is said to be the greatest common divisor, or factor, of 8 and 12: hence,

74. The Greatest Common Divisor, or factor, of two or more numbers is the greatest number that will divide each of them without a remainder.

REVIEW. - Define a unit. (1.) A number. (2.) A concrete number. (3.) A denominate number. (4.) An abstract number. (5.)

TO FIND THE GREATEST COMMON DIVISOR OF TWO OR

PROBLEM. 6, 12, and 18.

PROCESS. 6=3X2

12 = 3 × 2 × 2

18 = 3 × 2 × 3

MORE NUMBERS.

Find the greatest common divisor of

ANALYSIS.

- By inspection we find that 3 and 2 are the only common factors, or divisors, of 6, 12, and 18: hence the product of 3 and 2, or 6, is the greatest common divisor of 6, 12, and 18.

RULE. Resolve the numbers into their prime factors. The product of the factors which are common to all the numbers will be the greatest common divisor.

Find the greatest common divisors

When numbers cannot be easily factored, the pupil may use the following

RULE.-I. Divide the greater number by the less: if a remainder appears, divide the preceding divisor by it; and so continue to divide the last divisor by the last remainder, until nothing remains. The last divisor used is the greatest common divisor.

II. When there are more than two numbers, find the greatest common divisor of two of them, then of that divisor and one of the other numbers, &c. The last divisor will be the greatest common divisor.

REVIEW. Define notation. (7.) For what is the Roman notation used? (8.) What letters are used? What is the value of each? How does repeating a letter affect its value? How is 9 represented? How 11?