41. Mr. Gill, a drover, purchased 36 head of cattle, at 64 dollars a head, and 88 sheep, at 5 dollars a head: he sold the cattle for 40 dollars a head, and the sheep for 4 dollars apiece: did he make or lose, and how much?

42. Mrs. Louisa Wilsie has 3 houses, valued at 12530 dollars, 11324 dollars, and 9875 dollars: also a farm, worth 6720 dollars. She has a daughter and 2 sons. To the daughter she gives one-third the value of the houses and one-fourth the value of the farm, and then divides the remainder equally among the boys: how much did each receive?

43. Mr. Jones has a farm of 250 acres, worth 125 dollars per acre, and offers to exchange with Mr. Cushing, whose farm contains 185 acres, provided Mr. Cushing will pay him 20150 dollars difference: what was Mr. Cushing's farm valued at, per acre?

44. Mr. Sparks bought a third part of neighbor Spendthrift's farm for 2750 dollars: what would he have paid for the whole farm at the same rate?

45. George Wilson bought 24 barrels of pork, at 14 dollars a barrel; one-fourth of it proved damaged, and he sold it at half price, and the remainder he sold at an advance of 3 dollars a barrel: did he make or lose by the operation, and how much?

46. A gentleman, having 50000 dollars, spent half of it in buying 5 houses, which, after repairing at an expense of 1250 dollars, he sold at 6520 dollars each: what was his fortune after the transaction?

47. A gentleman bought 3 houses for 15850 dollars. For two he paid an equal price; and for the thirl, 850 dollars more than for either of the others: what did he pay for each?

48. Mr. J. Williams went into business with a capital of 25000 dollars in the first year he gained 2000; in the second year, 3500; in the third year, 4000 dollars: he then invested the whole in a cargo of tea and doubled his money: what was then the value of his fortune?

PROPERTIES OF NUMBERS.

Exact Divisors.

80. An EXACT DIVISOR of a number, is any number, except 1 and the number itself, that will divide it without a remainder. The dividend is then said to be divisible by the divisor.

81. An ODD NUMBER is not divisible by 2.

82. An EVEN NUMBER is one divisible by 2.

1. Three, is an exact divisor of any number, the sum of whose digits is divisible by 3.

2. Four, is an exact divisor of a number, when it will exactly divide the number expressed by the two right-hand digits.

3. Five, is an exact divisor of every number whose righthand figure is 0 or 5.

4. Six, is an exact divisor of any even number of which 3 is an exact divisor.

5. Nine, is an exact divisor of any number, the sum of whose digits is divisible by it.

6. Ten, is an exact divisor of every number whose righthand figure is 0.

83. A PRIME NUMBER is one which has no exact divisor: 1, 2, 3, 5, 7, 11, 13, 17, 19, &c.,

are prime numbers.

84. A COMPOSITE NUMBER is a number which has two or more exact divisors.

85. A FACTOR of a composite number, is any one of its exact divisors.

80. What is an exact divisor of any number? What is then said of the dividend?-81. What is an odd number?-82. What is an even number?-83. What is a prime number?-84. What is a composite number?-85. What is a factor?

CASE I.

86. To find the prime factors of a composite number. 1. What are the prime factors of 2310?

ANALYSIS.-We first divide by 2, the least prime factor of the given number. We then divide the quotient by 3, then the quotient by 5, and then by 7, when we obtain the quotient 11, which is prime. Hence, the prime factors of 2310 are, 2, 3, 5, 7, and 11. Hence, the following

Rule.

OPERATION

2) 2310

3) 1155

5) 385 7) 77

11

Divide the given number by any prime number that will exactly divide it: then divide the quotient in the same manner, and so on, till a quotient is found which is a prime number: the several divisors and the last quotient will be the prime factors.

Examples.

What are the prime factors of the following numbers?

87. To find the prime factors common to two or more composite numbers.

1. What are the common prime factors of 70, 210, and 280?

ANALYSIS.-It is plain that 2 is an exact divisor of all the numbers, and hence, a common factor: 5 is an exact divisor of the first set of quotients, 35, 105, and 140; hence, it is a common factor: 7 is an exact divisor of the second set of quotients; hence, it is a com

OPERATION.

2) 70. 210. 280

105 140

21.

5) 35
7) 7

28

1 3

4

mon factor, and the third set of quotients have no exact divisor. Hence, the following

Rule.

I. Write the numbers in a row, and then divide them by any prime number that is an exact divisor of all of them:

The

II. Divide each set of quotients in the same manner, until a set is found which has no exact divisor. divisors will be the common prime factors.

NOTE. The product of the prime factors, is the greatest factor common to all the numbers. Thus, 2 x 5 x 7 = 70, is the greatest factor common to 70, 210, 280.

Examples.

1. What are the prime factors common to 6, 9, and 24? 2. What are the prime factors common to 21, 63, and 84? 3. What are the prime factors common to 21, 63, and 105? 4. What are the common prime factors of 28, 42, and 70 ? 5. What are the prime common factors of 84, 126, and 210? 6. What are the prime factors of 210, 315, and 525?

Cancellation.

88. CANCELLATION is a process of shortening Arithmetical operations in Division, by omitting, or canceling, factors common to the dividend and divisor.

89. Cancellation depends upon the principle that,

If the dividend and divisor be both divided by the same number, the quotient will not be changed.

86. How do you find the factors of a composite number? 87. How do you find the prime factors common to two or more composite numbers? What is the greatest factor common to all of the numbers?

88. What is Cancellation?

89. On what principle does Cancellation depend?

1. Divide 63 by 21.

ANALYSIS.-Resolve the dividend and divisor into factors, then cancel those which are common, and mark the canceled fig

ures.

2. In 7 times 56, how many times 8?

ANALYSIS.-Resolve 56 into the two factors 7 and 8, and then cancel the 8.

56×7

8

3. In 36 times 15, how many times 45? ANALYSIS.-We see that 9 is a factor of 36 and 45. Divide by this factor, and write the quotient 4 over 36, and the quotient 5 below 45. Again, 5 is a factor of 15 and 5. Divide 15 by 5, and write the quotient 3 over 15, and the quotient of 5 by 5, under 5. Dividing 5 by 5, reduces the divisor to 1: hence, the 4 x 3 12 true quotient is, 1 1

- = 12.

90. Hence, for the operations of Cancellation, we have the following

Rule.

Cancel those factors that are common to the dividend and divisor, and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor.

NOTES.-1. If one of the numbers contains a factor equal to the product of two or more factors of the other, they may all be canceled.

2. If the product of two or more factors of the dividend is equal to the product of two or more factors of the divisor, they may all be canceled.

3. If all the factors of the dividend are canceled, the quotient 1 must be put for the factor last canceled.

90. What is the rule for the operations of Cancellation?