Ex. 1. A man sold 25 hundred weight of iron at 5 dollars per hundred weight, and expended the money for flour at 5 dollars barrel; how many barrels did he purchase?

per

OPERATION.

Dividend
Divisor

X 25
$

25.

Ans. 25 barrels.

We first indicate by their signs the multiplication and division reWe then, quired by the question. observing 5 to be a common factor

of the divisor and dividend, divide the divisor and dividend by this factor, or, which is the same thing, cancel or reject it in both, and obtain 25 for the quotient.

2. Divide the product of 12, 7, and 5, by the product of 5, 4, and 2.

Ans. 10.

Finding 4 in the divisor to be a factor of 12 in the dividend, we divide 12 by 4, cancelling these numbers, and use the 3 instead of 12. The factor 5, common to both dividend and divisor, having been cancelled, we divide the product of the remaining factors in the dividend by the product of those in the divisor, and obtain the quotient 103.

3. Divide the product of 8, 5, 16, and 21, by the product of 10, 4, 12, and 7.

Dividend

OPERATION.

4
5×16×21

Divisor IX 4 × 12 × 7

3

= 4, Quotient.

The product of the factors 8 and 5 in the dividend is equal to the product of 10 and 4 in the divisor; therefore we cancel these factors. Finding 16 in the dividend and 12 in the divisor may be divided by 4, they are cancelled, and use made of their quotients. Again, as the product of the factors 3 and 7 of the divisor equals the 28 of the dividend, we cancel them. The factor 4 alone remaining is the quotient.

QUESTIONS. How do you arrange the dividend and divisor for cancellation? How do you then proceed? Is the factor 5, in Ex. 1, reduced to 0 or 1 by being cancelled? How do you proceed when a number in the dividend and another in the divisor have a common factor? How do you proceed when the products of two or more factors in the dividend and divisor are alike?

RULE. -Cancel the factor or factors common to the dividend and divisor, and then divide the product of the factors remaining in the dividend by the product of those remaining in the divisor.

NOTE.-1. In arranging the numbers for cancellation, the dividend may be written above the divisor with a horizontal line between them, as in division (Art. 47); or, as some prefer, the dividend may be written on the right of the divisor, with a vertical line between them.

NOTE.-2. Cancelling a factor does not leave 0, but the quotient 1, to take its place, since rejecting a factor is the same as dividing by that factor (Art. 116). Therefore, for every factor cancelled, either in the dividend or divisor, the factor 1 remains.

EXAMPLES FOR PRACTICE.

4. Divide 42 × 19 by 19.

Ans. 42.

5. Divide the product of 8, 6, and 3, by the product of 6, 3, and 4. Ans. 2. 6. Divide the product of 17, 6, and 2, by the product of 6, 2, and 17.

Ans. 1.

7. Sold 15 pieces of shirting, and in each piece there were 30 yards, for which I received 10 cents per yard; expended the money for 10 pieces of calico, each containing 15 yards; what was the calico per yard? Ans. 30 cents. 8. Divide the product of 12, 7, and 5, by the product of 2, 4, and 3. Ans. 17. 9. Divide the product of 20, 13, and 9, by the product of 13, 16, and 1.

Ans. 11. 10. Divide the product of 9, 8, 2, and 14, by the product of 3, 4, 6, and 7.

Ans. 4. 11. Divide the product of 16, 5, 10, and 18, by the product of 8, 6, 2, and 12. Ans. 124.

12. Divide the product of 22, 9, 12, and 5, by the product of 3, 11, 6, and 4.

Ans. 15.

13. Divide the product of 25, 7, 14, and 36, by the product of 4, 10, 21, and 54. Ans. 113.

14. Divide the product of 26, 72, 81, and 12, by the product of 36, 13, 24, and 54.

Ans. 3. 15. Divide the product of 8, 5, 3, 16, and 28, by the product of 10, 4, 12, 4, and 7.

Ans. 4.

16. Divide the product of 8, 4, 9, 2, 12, 16, and 5, by the product of 4, 6, 6, 3, 8, 4, and 20.

Ans. 2.

17. Divide the product of 6, 15, 16, 24, 12, 21, and 27, by the product of 2, 10, 9, 8, 36, 7, and 81. Ans. 8.

QUESTIONS.-What is the rule for cancellation? How may the numbers be arranged for cancelling? What takes the place of a cancelled factor? What remains for every factor cancelled either in the dividend or divisor?

A COMMON DIVISOR.

ART. 118. A common divisor of two or more numbers is any number that will divide them without a remainder; thus, 2 is a common divisor of 2, 4, 6, and 8.

ART. 119. To find a common divisor of two or more numbers.

Ex. 1. What is the common divisor of 10, 15, and 25 ?

10 15

OPERATION.

=

=

5 X 2
5X3

25 = 5×5

RULE.

Ans. 5.

We resolve each of the given numbers into two factors, one of which is common to all of them. In the operation 5 is the common factor, and therefore must be a common divisor of the numbers.

- Resolve each of the given numbers into two factors one of which is common to all of them, and this common factor is a common divisor.

EXAMPLES FOR PRACTICE.

2. What is the common divisor of 3, 9, 18, 24 ? 3. What is the common divisor of 4, 12, 16, 28?

Ans. 3.

Ans. 2 or 4.

ART. 120. A divisor of any factor of a number is a divisor of the number itself. Thus 3, a divisor of 9, a factor of 45, is a divisor of 45 itself.

ART. 121. A common divisor of two numbers is a divisor of their sum and of their difference. Thus 4, a common divisor of 16 and 12, is a divisor of their sum, 28, and of their difference, 4.

ART. 122. A common divisor of the remainder and the divisor is a divisor of the dividend. Thus, in a division having 12 for remainder, 36 for divisor, and 48 for dividend, 12, a common divisor of the 12 and the 36, is also a divisor of the 48.

THE GREATEST COMMON DIVISOR.

ART. 123. The greatest common divisor of two or more numbers is the greatest number that will divide each of them without a remainder. Thus 6 is the greatest common divisor of 12, 18, and 24.

QUESTIONS. Art. 118. What is a common divisor of two or more numbers? - Art. 119. What is the rule? - Art. 121. Of what is the common divisor of two numbers a divisor? - Art. 122. Of what is a common divisor of the less of two numbers and of their difference a divisor?-Art. 123. What is the greatest common divisor of two or more numbers?

ART. 124. To find the greatest common divisor of two or more numbers.

Ex. 1. What is the greatest common divisor or measure of 84 and 132 ?

84 132

FIRST OPERATION.

=

2×2×3× 7 = 2 X 2 × 3 × 11 2 X2 X3=12.

Ans. 12. Resolving the numbers into their prime factors (Art. 114), thus, 84 = 2 × 2 × 3 × 7, and 132=2X 2 X 3 X 11, we find the factors 2 X2 X3 are common to both. Since only these common factors, or the product of two or more of such factors, will exactly divide both numbers, it follows that the product of all their common prime factors must be the greatest factor that will exactly divide both of them. Therefore 2 × 2 × 3 = the greatest common divisor required.

= = 12 is

The same result may be obtained by a sort of trial process, as by the second operation.

SECOND OPERATION.

84) 132 (1
84

48) 84 (1
48

36) 48 (1
36

It is evident, since 84 cannot be exactly divided by a number greater than itself, if it will also exactly divide 132, it will be the greatest common divisor sought. But, on trial, we find 84 will not exactly divide 132, there being a remainder, 48. Therefore 84 is not a common

12) 36 (3 divisor of the two numbers.

36

We know a common divisor of 48 and 84 will also be a divisor of 132 (Art. 122). We next try to find that divisor. It cannot be greater than 48. But 48 will not exactly divide 84, there being a remainder, 36; therefore 48 is not the greatest common divisor.

Again, as the common divisor of 36 and 48 will also be a divisor of 84 (Art. 122), we try to find that divisor, knowing that it cannot be greater than 36. But 36 will not exactly divide 48, there being a remainder, 12; therefore 36 is not the greatest common divisor.

As before, the common divisor of 12 and 36 will be a divisor of 48 (Art. 122); we make a trial to find that divisor, knowing that it cannot be greater than 12, and find 12 will exactly divide 36. Therefore 12 is the greatest common divisor required.

RULE 1.-Resolve the given numbers into their prime factors. The product of all the factors common to the several numbers will be the greatest common divisor. Or,

RULE 2.- Divide the greater number by the less, and if there be a

QUESTION. Art. 124. What are the rules for finding the greatest common divisor of two or more numbers?

remainder divide the preceding divisor by it, and so continue dividing until nothing remains. The last divisor will be the greatest common divisor.

NOTE. When the greatest common divisor is required of more than two numbers, find it of two of them, and then of that common divisor and of one of the other numbers, and so on for all the given numbers. The last common divisor will be the greatest common divisor required.

and 168?
Ans. 24.
and 121?
Ans. 1.
18, 24, and
Ans. 6.

5. What is the greatest common divisor of 12,

30?

6. Having three rooms, the first 12 feet wide, the second 15 feet, and the third 18 feet, I wish to purchase a roll of the widest carpeting that will exactly fit each room without any cutting as to width. How wide must it be? Ans. 3 feet.

A COMMON MULTIPLE

ART. 125. A multiple of a number is a number that can be divided by it without a remainder; thus 6 is a multiple of 3.

ART. 126. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus 12 is a common multiple of 3 and 4.

ART. 127. The least common multiple of two or more numbers is the least number that can be divided by each of them without a remainder; thus 30 is the least common multiple of 10 and 15.

NOTE. A multiple of a number contains all the prime factors of that number; and the common multiple of two or more numbers contains all the prime factors of each of the numbers. Therefore, the least common multiple of two or more numbers must be the least number that will contain all the prime factors of them, and none others. Hence it will have each prime factor taken only the greatest number of times it is found in any of the several numbers.

QUESTIONS.

- Art. 125. What is a multiple of a number? What is the least common multiple of a number?