GENERAL PRINCIPLES OF DIVISION.

86. The quotient in Division depends upon the relative values of the dividend and divisor. Hence any change in the value of either dividend or divisor must produce a change in the value of the quotient. But some changes may be produced upon both dividend and divisor, at the same time, that will not affect the quotient. The laws which govern these changes are called General Principles of Division, which we will now examine.

I. 5496.

Multiplying the dividend by 3, we have

54 × 39162 ÷ 9=

18,

and 18 equals the quotient, 6, multiplied by 3. Hence, Multiplying the dividend by any number, multiplies the quotient by the same number.

II. Using the same example, 54 ÷ 9 = 6.

Dividing the dividend by 3 we have

54918÷÷9=2,

and 2 the quotient, 6, divided by 3. Hence, Dividing the dividend by any number, divides the quotient by the same number.

III. Multiplying the divisor by 3, we have

54 ÷ 9 × 3 = 54 ÷ 27 = 2,

and 2 the quotient, 6, divided by 3. Hence, Multiplying the divisor by any number, divides the quotient by the same number.

IV. Dividing the divisor by 3, we have

5454÷ 3 = 18,

Upon what does the value of the quotient depend? What is the first general principle of division? Second? Third? Fourth?

GENERAL PRINCIPLES OF DIVISION. 65.

and 18 the quotient, 6, multiplied by 3. Hence, Dividing the divisor by any number, multiplies the quotient by the same number.

V. Multiplying both dividend and divisor by 3, we have

54 x 39 x 3 = 162 ÷ 27 = 6.

Hence, Multiplying both dividend and divisor by the same number, does not alter the value of the quotient.

VI. Dividing both dividend and divisor by 3, we have

Hence, Dividing both dividend and divisor by the same number, does not alter the value of the quotient.

87. These six examples illustrate all the different changes we ever have occasion to make upon the dividend and divisor in practical arithmetic. The principles upon which these changes are based may be stated as follows:

PRIN. I. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient. (86. I and II.) PRIN. II. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient. (86. III and IV.)

PRIN. III. Multiplying or dividing both dividend and divisor by the same number, does not alter the quotient. (86. V and VI.)

88. These three principles may be embraced in one

GENERAL LAW.

A change in the DIVIDEND produces a LIKE change in the quotient; but a change in the DIVISOR produces an OPPOSITE change in the quotient.

I a number be multiplied and the product divided by the same number, the quotient will be equal to the number multiplied. Thus, 15 x 460, and 60÷4=15.

Fifth? Sixth? Into how many general principles can these br condensed? What is the first? Second? Third? In what genera law are these embraced?

EXACT DIVISORS.

89. An Exact Divisor of a number is one that gives a whole number for a quotient.

As it is frequently desirable to know if a number has an exact divisor, we will present a few directions that will be of assistance, particularly in finding exact divisors of large numbers.

A number whose unit figure is 0, 2, 4, 6, or 8, is called an Even Number. And a number whose unit figure is 1, 3, 5, 7, or 9, is called an Odd Number.

2 is an exact divisor of all even numbers.

4 is an exact divisor when it will exactly divide the tens and units of a number. Thus, 4 is an exact divisor of 268, 756, 1284.

5 is an exact divisor of every number whose unit figure is 0 or 5. Thus, 5 is an exact divisor of 20, 955, and 2840.

8 is an exact divisor when it will exactly divide the hundreds, tens, and units of a number. Thus, 8 is an exact divisor of 1728, 5280, and 213560.

9 is an exact divisor when it will exactly divide the sum of the digits of a number. Thus, in 2486790, the sum of the digits 2+4+8+6+7+9+0=36, and 36÷÷9=4.

10 is an exact divisor when 0 occupies units' place. 100 when 00 occupy the places of units and tens.

1000 when 000 occupy the places of units, tens, and hundreds, etc.

A composite number is an exact divisor of any number, when all its factors are exact divisors of the same number. Thus, 2, 2, and 3 are exact divisors of 12; and so also are 4 (=2×2) and 6 (=2×3).

An even number is not an exact divisor of an odd number. If an odd number is an exact divisor of an even number,

What is an exact divisor? What is an even divisor? An odd number? When is 2 an exact divisor? 4? 5? 9? 10? 100? 1000? When is a composite number an exact divisor? An even number is not an exact divisor of what? An odd number is an exact divisor of what?

twice that odd number is also an exact divisor of the even number. Thus, 7 is an exact divisor of 42; so also is 7 × 2, or 14.

PRIME NUMBERS.

90. A Prime Number is one that can not be resolved or separated into two or more integral factors.

For reference, and to aid in determining the prime factors of composite numbers, we give the following:

91. To resolve any composite number into its prime factors.

What is a prime number? In factoring numbers, Case I is what?

1 What are the prime factors of 2772 ?

ANALYSIS. Divide the given number by 2, the least prime factor, and the result by 2; this gives an odd number for a quotient, divisible by the prime factor, 3, and the quotient resulting from this division is also divisible by 3. The next quotient, 77, we divide by its least prime factor, 7, and obtain the quotient 11; this being a prime number, the division can not be carried further. The divisors and last quotient, 2, 2, 3, 3,7, and 11 are all the prime factors of the given number, 2772.

RULE. Divide the given number by any prime factor; divide the quotient in the same manner, and so continue the division until the quotient is a prime number. The several divisors and the last quotient will be the prime factors required.

PROOF. The product of all the prime factors will be the given number.

EXAMPLES FOR PRACTICE.

2. What are the prime factors of 1140? Ans. 2, 2, 3, 5, 19.

3. What are the prime factors of 29925?

4. What are the prime factors of 2431?

5. Find the prime factors of 12673.

6. Find the prime factors of 2310.

7. Find the prime factors of 2205.

8. What are the prime factors of 13981 ?

CASE II.

92. To resolve a number into all the different sets of factors possible.

1. In 36 how many sets of factors, and what are they?

Give explanation. Rule. Proof. Case II is what?