29. Divide 4365 by 250; by 1663. 30. Divide 15300 by 163; by 3331.

50. PRINCIPLE.—The multiplying of both divisor and dividend by the same number does not change the value of the quotient.

51. RULE.-To divide by a convenient part of 10, 100, 1000, etc., Multiply the dividend by the number denoting how many times the divisor is contained in 10, or 100, or 1000, etc., and divide the product by 10, or 100, or 1000, etc.

Case IV.

The Divisor a Composite Number.

31. Divide 18315 by 45.

Since 45: 5 × 9, the quotient obtained by dividing 18315 by 5, is 9 times too large, and hence this quotient (3663) divided by 9, is the true quotient.

The process of dividing by the factors of the divisor successively is the same in principle as the division of both dividend and divisor by these factors successively, which (Art. 48) does not change the value of the quotient. See "Illustrative Process."

A unit of the first quotient equals 2 units of the dividend, and hence the second remainder (3) equals 3 × 2 units of the dividend.

A unit of the second quotient equals 8 units of the first quotient, and hence the third remainder (2) equals 2 X 8 units of the second quotient = 2 X 8 X 2 units of the dividend. Hence the first remainder is 1, the second 6; the third 32; and the total, or true remainder, 39.

NOTE. The teacher can illustrate this process by considering the dividend (3687) pints. The first quotient will be quarts, the second pecks, and third bushels, and the first remainders will be 1 pt., the second, 3 qt., and the third, 2 pk. 1 pt. + 3 qt. + 2 pk. = 39 pt.

36. Divide 34567 by 63; by 72.

37. Divide 120473 by 56;

by 81.

38. Divide 400671 by 64; by 77. 39. Divide 346000 by 55; by 96. 40. Divide 47633 by 90; by 110.

52. PRINCIPLE.-The division of both divisor and dividend by the same number does not change the value of the quotient.

53. RULE. To divide by a composite number, 1. Resolve the divisor into convenient factors; divide the dividend by one of these factors, the quotient thus obtained by another, and so on until all the factors are used as divisors. The last quotient will be the true quotient.

2. Multiply each remainder, except the first, by all the divisors preceding its own. The sum of these products and the first remainder will be the true remainder.

SECTION VII.

PROPERTIES OF NUMBERS.

PRIME AND COMPOSITE NUMBERS AND FACTORS.

NOTE. The terms number, divisor, and factor, used in this section, denote integral numbers.

1. What two numbers besides itself and 1 will exactly divide 10? 21? 35? 63? 77?

2. What numbers besides itself and 1 will exactly divide 7? 11? 17? 23? 37? 41 ?

3. What numbers will exactly divide 15? 13? 28? 29? 42? 43?

NOTE. Since every integer is exactly divisible by itself and 1, these divisors need not be given.

4. What numbers will exactly divide 30? 31? 45? 53? 56? 67? 65 ?

5. Name all the prime numbers between 0 and 20; 30 and 50.

6. Name all the composite numbers between 20 and 30; 50 and 70.

7. What are the prime divisors of 6? 15? 18? 21? 30? 45? 50? 54?

8. What are the prime factors of 12? 24? 35? 39? 42? 9. What are the prime factors of 27? 36? 49? 56? 63? 66? 72? 84?

10. Of what numbers are 2 and 5 prime factors? 2, 3, and 5? 2, 5, and 7? 3, 5, and 7?

11. Of what numbers are 2, 2, and 3 prime factors? 2, 3, 3, and 5? 2, 3, 5, and 7?

12. What prime factor is common to 9 and 12? 15 and 25? 18 and 30? 21 and 28?

13. What prime factor is common to 24 and 27? 35 and 42? 44 and 77? 35 and 50? 63 and 70?

DEFINITIONS, PRINCIPLES, AND RULES.

54. The Divisor of a number is any number that will exactly divide it.

55. Numbers are either Prime or Composite.

A Prime Number has no divisor except itself and

one.

A Composite Number has other divisors besides itself and one.

Every composite number is the product of two or more numbers, called factors.

56. Two or more numbers are prime to each other, or relatively prime, when they have no common divisor except 1. Thus, 9 and 16 are prime to each other.

All prime numbers are prime to each other. Composite numbers may be relatively prime, as 9 and 10; 16 and 25.

57. A Factor of a number is its divisor.

A Prime Factor of a number is its prime divisor.

The terms divisor and factor differ only in their use, the former implying division and the latter multiplication. A divisor or factor of a number is also called its measure.

58. When a number is a factor of each of two or more numbers, it is called their Common Factor.

is a common factor of 15 and 20.

59. Whole numbers are either Even or Odd.

Thus, 5

An Even Number is exactly divisible by 2; as, 2, 4, 6, 8, 10, 12, etc.

An Odd Number is not exactly divisible by 2; as, 1, 3, 5, 7, 9, 11, 13, etc.

All the even numbers except 2 are composite. Some of the odd numbers are composite and others are prime.

60. PRINCIPLES.-1. A factor of a number is a factor of any number of times that number.

2. A common factor of two or more numbers is a factor of their sum.

3. A composite number is the product of all its prime factors. 4. If a composite number composed of two factors be divided by one factor, the quotient will be the other factor.

5. If any composite number be divided by a factor, or by the product of any number of its factors, the quotient will be the product of the remaining factors.

61. RULES.-1. To resolve a composite number into its prime factors, Divide it by any prime divisor, and the quotient by any prime divisor, and so continue until a quotient is obtained which is a prime number. The several divisors and

the last quotient are the prime factors.

2. To find the common factors of two or more numbers, Resolve the given numbers into their prime factors and select the factors which are found in all the numbers.

CANCELLATION.

33. Divide the product of 4, 7, 9, and 12 by the product of 4, 7, and 9.

the dividend. Since dividing both dividend and divisor by the same number does not affect the value of the quotient (Art. 48), divide each by 4, 7, and 9. This may be done by canceling, as indicated in the process. The quotient is 12.

34. Multiply 4 X 7 by 12, and divide the product by 4 times 12.

35. Divide 6 × 8 × 20 by 4 × 20.

36. Divide 5 X 7 X 11 X 13 by 7 × 131.