61. If 216 pianos cost $112320, what will 519 cost?

62. How many pounds of coffee, at 38 cents a pound, will pay for 2 hogsheads of sugar containing 1160 pounds each, at 19 cents a pound?

63. A farmer having $ 3038, bought 15 tons of hay at $11 per ton, 3 yoke of oxen at $155 each, 375 sheep at $5 each, and spent the rest for cows at $41 apiece. How many cows did he buy?

64. A man in his will gave to each of his 2 sons $7600; to a third son $1500; to each of 3 daughters $3775, and the balance $6877 to his wife. His wife died, however, and the whole property was divided equally among his children; what did each receive?

65. The expenses of a picnic party of 9 gentlemen and 8 ladies were $2.40 each. The gentlemen paid all the expenses. What did each pay?

66. Sold 160 tons of coal at $5 per ton, and a number of tons at $3 per ton. The value of all the coal sold was $965. How many tons were there?

67. Bought 960 acres of land for $12000. Sold of it at $12 per acre, of it at $15 per acre, and the remainder for $20 per acre. Did I gain or lose, and how much?

REVIEW QUESTIONS.

2. What is a number? 21. What is an integer? 55. What is a concrete number? 56. An abstract number?

4. What is notation? 5. Numeration? 18. What are principles of notation? 31. What is addition? 37. What are principles of addition? 41. What is subtraction? 46. What are principles of subtraction?

50. What is multiplication? 58. What are principles of multiplication? 63. What is division? 71. What are principles of division ?

FACTORS.

77. 1. What two numbers multiplied together produce 2? 3? 5?

2. What numbers, other than itself and 1, will divide 6 without a remainder?

3. What number multiplied by 2 will produce 6?

4. Name all the numbers, other than itself and 1, which will divide 21 without a remainder.

5. What number multiplied by 3 will produce 21 ?

6. Name the numbers between 1 and 15 which are the product of two or more numbers greater than 1.

78. The Factors of a number are the integers which produce the number when multiplied together.

Thus,

3 and 5 are factors of 15, and 2, 3, and 3 are factors of 18.

79. A Composite Number is a number having other factors than itself and one.

NOTE.

Thus,

4, 6, 8, 9, and 10 are composite numbers.

One number is said to be divisible by another when there is no remainder in the division. Thus,

A composite number is divisible by any of its factors.

80. A Prime Number is a number having no other factors than itself and one. Thus,

1, 2, 3, 5, 7, 11 are prime numbers.

81. A Prime Factor is a factor that is a prime number.

NOTE. As every integer is a factor of itself, and has one as the other factor, in speaking of the factors of a number we usually exclude the number itself and one.

82. Every prime number, except 2 and 5, has 1, 3, 7, or 9 for its unit figure.

NOTE. — 2 is a factor of any number ending in 0, or whose ones are divisible by 2. 3 or 9 is a factor of any number the sum of whose figures is divisible by 3 or 9. 5 is a factor of any number ending in 0 or 5.

8. What are the prime factors of 84? Of 144? Of 160? 9. What are the prime factors of 462? Of 576? Of 1008?

83. Rule for finding the Prime Factors of a Number. Divide the given number by any prime number above one that will exactly divide it. Divide the quotient, if composite, in like manner, and so continue until a prime quotient is found. The several divisors and the last quotient will be the prime factors.

22. What is the largest prime factor of 1184?

23. Find the prime factors of 4389.

24. Find the prime factors of 6300.

25. What prime numbers multiplied together produce 40579?

CANCELLATION.

26. What is the quotient of 63 divided by 21?

27. What is one seventh of 63 divided by 4 of 21 ?

28. What is the quotient of of 63, divided by of 21? 29. What is the quotient of 5 X 3 X 2, divided by 2 × 3? Of 3 × 5, divided by 3? Of 2 × 5, divided by 2?

84. Cancellation is striking out the same factor from both dividend and divisor.

85. Principles of Cancellation.

1. Striking out a factor of a number divides the number by the factor.

2. Striking out the same factors from dividend and divisor does not affect the quotient.

and divisor the factors 16 and 3, common to both, there remains only the factor 5 in the dividend, which is the required quotient.

31. Divide 11 times 1476 by 6 times 132.

32. Divide 45 x 20 x 7 by 49 x 4 x 9.

33. (54 × 3 × 4 × 15) ÷ (18 × 12 × 10) = ?

86. Rule for Cancellation.

Strike out all the factors common to both dividend and divisor, and then divide as may be required.

34. Divide 255 × 63 × 4 by 340 × 12 × 7.

35. Divide 18 × 16 × 10 × 5 by 12 × 8 × 6 × 2.

36. Divide 50 × 36 × 14 by 54 × 10 × 4 × 3.

37. Divide 545 × 105 × 11 by 35 × 33 × 5.

38. Multiply 64 by 63 and divide the product by 168.

39. What is the quotient of 36 × 21 × 14 divided by 27 X 7 X 6?

40. How many tons of hay at $24 a ton must be given for 4 cows at $42 each?

41. How many yards of cloth can be bought for $95, when 24 yards can be bought for $120?

42. If 100 men can perform a piece of work in 12 days, in how many days can 150 men perform it?

43. A fort has provisions for 225 men 12 months. How long will it last 675 men?

44. Exchanged 15 pieces of muslin, each containing 30 yards at 10 cents a yard, for 3 pieces of flannel, each containing 50 yards. What was the flannel a yard ?

GREATEST COMMON DIVISOR.

45. What number will divide both 21 and 35, without a remainder?

46. What factor is common to 21 and 35? To 15 and 25 ? 47. What is the greatest number that will divide both 18 and 30, without a remainder?

48. What is the greatest factor common to 18 and 30? To 16 and 24?

87. A Common Divisor of two or more numbers is any factor found in each of them. Thus,

7 is a common divisor of 14 and 21.