If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend,

Half that divisor is contained twice as many times; A third of that divisor, three times as many times, &c. Thus, 6 is contained in 24, 4 times; 6÷2 or 3 (half of 6,) is contained in 24, 8 times; (i. e. twice 4 times;) 6-3 or 2, (a third of 6,) is contained in 24, 12 times; (i. e. three times 4 times ;) &c. Hence,

86. If the dividend remains the same, dividing the divisor by any number, is in effect multiplying the quotient by that number.

87. From the preceding articles, it is evident that any given divisor is contained in any given dividend just as many times as twice that divisor is contained in twice that, dividend; three times that divisor in three times that dividend, &c.

Conversely, any given divisor is contained in any given dividend just as many times as half that divisor is contained in half that dividend; a third of that divisor in a third of that dividend, &c.

Thus, 4 is contained in 12, 3 times;

2 times 4 is contained in 2 times 12, 3 times;
3 times 4 is contained in 3 times 12, 3 times, &c.

Again, 6 is contained in 24, 4 times;

62 is contained in 24-2, 4 times;

63 is contained in 24÷3, 4 times, &c. Hence,

88. If the divisor and the dividend are both multiplied, or both divided by the same number, the quotient will not be altered.

89. If any given number is multiplied and the product divided by the same number, its value will not be altered. Thus 12×5=60; and 60÷÷5=12.

QUEST.-86. What of dividing the divisor? 88. What is the effect upon the quotient if the divisor and dividend are both multiplied or both divided by the same number? 89. What is the effect of multiplying and dividing any given number by the same number?

CANCELATION.*

90. We have seen that division is finding a quotient, which, multiplied into the divisor will produce the dividend. (Art. 65.) If, therefore, the dividend is resolved into two such factors that one of them is the divisor, the other factor will, of course, be the quotient. Suppose, for example, 42 is to be divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Hence,

Canceling a factor of any number, divides the number by that factor.

OBS. The term cancel means to erase or reject.

91. When the dividend is the product of two or more factors, one of which is the same as the divisor, the division may be performed by CANCELING that factor in the divisor and dividend. (Art. 89.)

21. Divide the product of 19 into 25 by 19..

Common Method.

19 25

95 38

19)475(25 Ans.

38

95

95

By Cancelation.
19)19×25

25 Ans.

Cancel the factor 19, which is common both to the divisor and dividend, and 25, the other factor of the dividend, will be the quotient. (Art. 90.)

Suggestion. 28-4×7. We may therefore contract

QUEST.-90. What is the effect of canceling a factor of any number? Obs. What is meant by the term cancel? 91. When the divisor is a factor of the dividend, how may the division be performed?

* Birk's Arithmetical Collections, London, 1764.

the division by canceling the 7, which is a factor both of the dividend and the divisor. (Arts. 88, 90.)

Operation. 7)63X4×7 252 Ans.

The product of 63 × 4, the other factors of the dividend, is the quotient.

26. In 32 times 84, how many times 8? Ans. 336. 27. In 35 times 95, how many times 7? 28. In 48 times 133, how many times 8? 29. In 96 times 156, how many times 12 ? 30. Divide 168×2×7 by 7×3.

Operation. 7X3)168×2×7 3)336

112 Ans.

We cancel the factor 7, which is common to the divisor and dividend, then divide the product of 168 into 2 by 3.

31. Divide the product of 8, 6, and 12, by the product of 2, 6, and 8.

Solution. 2×6×8)8×6×12=6 Ans.

Note.-We cancel the factors 2 and 6 in the divisor and the 12 in the dividend, &c. Canceling the same factor both in the divisor and dividend, is in effect dividing them both by the same number, and consequently does not effect the quotient. (Art. 88.) Hence,

91. a. When the divisor and dividend have factors common to both, the division may be performed by canceling the common factors, and then dividing those that are left as before.

32. Divide the product of 7, 9, 15, and 8 by the product of 5, 7, and 8.

33. Divide the product of 6, 3, 7, and 4 by the product of 12 and 6.

34. Divide the product of 2, 28, and 15 by 30. 35. Divide the product of 5, 6, and 56 by 7×8.

Note. This method of contracting arithmetical operations, is called Cancelation. It applies with great advantage to that class of examples and problems which involve both multiplication and division; that is, when the product of two or more numbers is to be divided by another number, or by the product of two or more numbers. Its further developments and application may be seen in reduction of compound fractions to simple ones; in multiplication and division of fractions; in simple and compound proportion, &c., &c.

GREATEST COMMON DIVISOR.

92. A Common Divisor of two or more numbers, is a number which will divide them without a remainder. Thus 2 is a common divisor of 4, 6, 8, 12, 16, &c.

93. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide them without a remainder. Thus 6 is the greatest common divisor of 12, 18, and 24.

OBS. 1. One number is said to be a measure of another, when the former is contained in the latter any number of times without a remainder. Hence, a Com. divisor is often called a Common Measure.

2. It will be seen that a common divisor of two or more numbers, is simply a factor which is common to those numbers, and the greatest common divisor is the greatest factor common to them. Hence,

94. To find a common divisor of two or more numbers.

Resolve each number into two or more factors, one of which shall be common to all the given numbers.

Ex. 1. Find a common divisor of 8, 10, and 12.

Solution. 8 may be resolved into the factors 2 and 4; that is, 8=2×4; 10=2×5; and 12=2×6. The fac tor 2 is common to each number and is therefore a common divisor of them.

2. Find a common divisor of 9, 15, 18, and 24.

OBS. The following facts may assist the learner in finding common divisors:

1. Any number ending in 0, or an even number, as 2, 4, 6, &c. may be divided by 2.

2. Any number ending in 5 or 0, may be divided by 5.

3. Any number ending in 0, may be divided by 10.

4. When the two right hand figures are divisible by 4, the whole number may be divided by 4.

3. Find a common divisor of 16, 20, and 36.

QUEST.-92. What is a common divisor of two or more numbers? 93. What is the greatest common divisor of two or more numbers? Obs. When is one number said to be a measure of another? What is a common divisor sometimes called? 94. How do you find a common divisor of two or more numbers?

4. Find a common divisor of 35, 50, 75, and 80. 5. Find a common divisor of 148 and 184.

6. Find a common divisor of 126 and 4653.

95. No two numbers which have not a common factor, can have a common divisor greater than a unit. 'Thus the factors of 8 are 2 and 4; the factors of 15 are 3 and 5; hence, 8 and 15 have no common divisor.

96. To find the greatest common divisor of two numbers.

Divide the greater number by the less; then the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor.

7. What is the greatest common divisor of 70 and 84 ?

Operation.
70)84(1
70

14)70(5
70

Dividing 84 by 70, the remainder is 14; then dividing 70 (the preceding divisor) by 14, (the last remainder,) nothing remains. Hence, 14, the last divisor, is the greatest common divisor.

8. What is the greatest common divisor of 63 and 147? 9. What is the greatest common divisor of 91 and 117? 10. What is the greatest common divisor of 247 and 323?

11. What is the greatest common divisor of 285 and 465 ?

12. What is the greatest common divisor of 2145 and 3471 ?

97. To find the greatest common divisor of more than two numbers.

First find the greatest common divisor of any two of them; then, that of the common divisor thus obtained and of another given number, and so on through all the given

QUEST.-95. If two numbers have not a common factor, what is true as to a common divisor? 96. How find the greatest common divisor of two numbers? 97. Of more than two?