sor and quotient are the two factors, which, being multiplied together, produce the dividend (Art. 50), it is plain, if we cancel the factor 5, thus 5 × 3 = 15, the remaining factor 3 is the quotient, and the dividend 15 has been divided by 5.

ART. 115. Method of cancelling, when there is one factor, or more, common to the dividend and divisor.

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 per barrel; how many barrels did he purchase?

OPERATION.

Dividend × 25

Divisor

5

Ans. 25 barrels.

The price of the iron per hundred 25 weight must be multiplied by the number of hundred weight sold, to obtain the value of the whole; and this product, being divided by the cost of the flour per barrel, will give the number of barrels bought. But we may indicate this multiplication and division by their signs, without actually performing the operations. In this example, 5 is a common factor of the divisor and dividend; therefore, we divide the divisor and dividend by this factor, or, which is the same thing, cancel it in both, and obtain 25 for the quotient.

2. Divide the product of 15, 2, 4, and 6, by the product of 4, 3, 2, and 5, and find the quotient. Ans. 6.

In this example we cancel the common factors in the divisor and dividend, and divide the product of the remaining factors in the dividend by the product of those in the divisor, and obtain the quotient 6.

RULE. 1. Write the dividend above the divisor, with a horizontal line between them, as in division. (Art. 47.)

2. Cancel the factor or factors common to the dividend and divisor, and, if there is only one factor remaining in the dividend and one

QUESTIONS. Art. 115. 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? If all the factors, both in the dividend and divisor, are cancelled, what will the quotient be? If more than one factor remain in the dividend and divisor after cancelling, how do you obtain the quotient? What is the rule for cancellation?

in the divisor, divide the factor of the dividend by the factor of the divisor.

3. If there are two or more factors remaining in the dividend, and two or more in the divisor, divide the product of the factors of the dividend by the product of the factors of the divisor.

NOTE. When a factor is cancelled, it is not reduced to 0, but to unity, or 1. Therefore, when all the factors are cancelled, either in the dividend or divisor, the factor 1 remains, and must be used as a factor of the divisor or dividend, as the case may be.

EXAMPLES FOR PRACTICE.

3. Divide 27 × 16 by 27.

4. Divide 42 x 19 by 19.

Ans. 16.

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.

ART. 116. When a number in the dividend and another in the divisor have a common factor.

8. Divide the product of 12, 7, and 5, by the product of 2, 4, and 3. Ans. 171.

It will be seen, in this example, that 4 in the divisor is a factor of 12 in the dividend. Therefore we divide 12 by 4, cancelling these numbers, and use the quotient 3 instead of 12. The operation may be carried still further by cancelling this factor 3, and a similar one in the divisor.

9. Divide the product of 20, 13, and 9, by the product of 13, 16, and 1. Ans. 111.

QUESTIONS.

Art. 116. How do you proceed when a number in the dividend and another in the divisor have a common factor? Is the common factor always one of the two numbers ?

In this example, 20 in the dividend and 16 in the divisor may be divided by 4. We therefore cancel these numbers and use their quotients in the operation.

RULE. When there is a factor in the dividend and another in the divisor which may be divided by the smaller factor, or by some other number, without a remainder, cancel these factors, and use the quotients arising from the division instead of them.

EXAMPLES FOR PRACTICE.

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

Ans. 124.

Ans. 15. 13. Divide the product of 25, 7, 14, and 36, by the product of 4, 10, 21, and 54.

Ans. 117.

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

Ans. 3.,

ART. 117. When the product of two or more factors of the dividend is equal to the product of two or more factors of the divisor, or conversely.

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

Dividend
Divisor

OPERATION.

X5 × 3 × 16 × 28 10 × 4 × 12 × 4 × 7

= 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. Again, the product of the factors 3 and 16 is equal to the product of

QUESTIONS. What is the rule for cancelling when a number in the dividend and another in the divisor have a common factor? - Art. 117. How do you proceed when the products of two or more factors in the dividend and divisor are alike?

the factors 12 and 4, which we also cancel, and there remains 28 divided by 7, which is equal to 4.

RULE. When the product of two or more factors in the dividend is equal to the product of two or more factors in the divisor, cancel these factors in both.

NOTE. If the product of two or more factors in the dividend is equal to any one factor in the divisor, or conversely, these factors may be cancelled in both.

EXAMPLES FOR PRACTICE.

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.

§ XVIII. PROPERTIES AND RELATIONS OF NUMBERS.

ART. 118. ALL numbers are either odd or even.

An odd number is a number that cannot be divided by 2 without a remainder; thus, 3, 7, 11.

An even number is a number that can be divided by 2 without a remainder; thus, 4, 8, 12.

Numbers are also either prime or composite.

A prime number is a number which can be divided only by itself or a unit; as 1, 3, 5, 7.

Numbers are said to be prime to each other when no number greater than a unit will divide them without a remainder; thus, 7 and 11 are prime to each other.

NOTE. For definition of composite numbers, see Art. 41.

ART. 119. A prime factor of a number is a factor which can be divided only by itself or a unit; thus, the prime factors of 21 are 3 and 7.

If the product of proceed? — Art.

What an even

QUESTIONS.-What is the rule for cases of this kind? two or more factors is equal to any one factor, how do you 118. What are all numbers? What is an odd number? number? What other distinctions of numbers are mentioned? prime number? When are numbers prime to each other? posite number? Art. 119. What is a prime factor of a number?

What is a What is a com

NOTE.-Unity or 1 is sometimes regarded as a prime factor; but since multiplying any number by 1 does not alter its value, we shall omit it when speaking of the prime factors of numbers.

ART. 120. To find the prime factors of a number.

Ex. 1. It is required to find the prime factors of 24.

OPERATION.

224

212

2 6

3

2 × 2 × 2 × 3 = 24

Ans. 2, 2, 2, 3.

Since 24 is not a prime number, we divide it by 2, the least prime number greater than 1, and obtain the quotient 12. And since 12 is a composite number, we divide this also by 2, and obtain a quotient 6. We next divide 6 by 2, and obtain 3 for a quotient, which is a prime number, and therefore the division ends. The several divisors and the last quotient constitute all the prime factors of 24.

RULE. — Divide the given number by the least prime number, greater than 1, that will divide it without a remainder, and then this quotient, if a composite number, in the same manner, and thus continue the division until a prime number is obtained for a quotient. The several divisors and the last quotient are its prime factors.

NOTE. The composite factors of any number may be found by multiplying together two or more of its prime factors.

EXAMPLES FOR PRACTICE.

2. What are the prime factors of 36? 3. What are the prime factors of 48? 4. What are the prime factors of 56 ? 5. What are the prime factors of 144?

Ans. 2, 2, 3, 3. Ans. 2, 2, 2, 2, 3. Ans. 2, 2, 2, 7.

Ans. 2, 2, 2, 2, 3, 3.

6. What are the prime factors of 3420?

Ans. 2, 2, 3, 3, 5, 19.

7. What are the prime factors of 18500 ?

Ans. 2, 2, 5, 5, 5, 37.

A COMMON DIVISOR.

ART. 121. A common divisor of two or more numbers

QUESTIONS.Is 1 usually considered a factor when speaking of the prime factors of a number?. -Art. 120. What is the rule for finding the prime factors of a number? How can the composite factors of a number be found? - Art. 121. What is a common divisor of two or more numbers ?