6. 11350 by 81.
7. 1130 by 54.

Ans. 24528.

Ans. 61020.

Ans. 2131570.

4. 4031 by 63. Ans. 253953.

8. Multiply 30451 by 70, or by 10 × 7.

Here, we annex a cipher to the multiplicand for the product of it by 10 (Art. 54), and then multiply by the 7; that is, we annex the cipher to the multiplicand, and multiply by the number expressed by the significant figure.

9. Multiply 40067 by 50, or by 10 X 5.

DIVISION BY FACTORS.

118. 1. Divide 15820 by 35, using factors.

OPERATION.

5) 15820
7) 3164

452

Ans. 2003350.

Since 35 times a number is equal to 7 times 5 times the number (Art. 117), one thirty-fifth of the number must equal one seventh of one fifth of the number.

One fifth of 15820 is 3164, and one seventh of one fifth of 15820, or one seventh of 3164, is 452.

Therefore, 15820 divided by 35 is equal to 452.

Had the divisor contained any other convenient set of factors than 5 and 7, they could have been used with like result.

2. Divide 6103 by 15, using factors.

OPERATION.

3) 6103

Dividing by the factor 3, we obtain 2034 threes, and a remainder of 1 unit.

5) 2034, Rem. 1 unit = 1 Dividing by the factor 5,

we obtain 406 (five times threes, or fifteens), and a remainder of 4 threes. 1 unit 4 threes, or 13,

What is 35 times a number equal to ?

RULE. Separate the divisor into convenient factors.

Divide the dividend by one of these factors, and the quotient by another, and so on, until all the factors have been used. The last quotient will be the one required.

Should there be one or more remainders, multiply each remainder by the divisors, if any, preceding the one that produced it, and the sum of the products plus the remainder left by the first division, if any, will be the true remainder.

119. Since dividing both the dividend and divisor by the same number will not change the quotient (Art. 73), it is often possible to shorten arithmetical operations by rejecting equal factors from both dividend and divisor, and using only the remaining factors.

The process has been called CANCELLATION, from the rejected factors being usually noted by being crossed or canceled. 11. Divide 3 times 90 by 54.

and 3, common to both dividend and divisor, we have §, or 5, which is the quotient.

Repeat the Rule. How is it often possible to shorten arithmetical operations? What is the process of rejecting factors called?

12. Divide 16 times 21 by 8 times 14.

OPERATION.

2

3

16 × 21

= 3, Ans.

8 × 14

7

1

Canceling the factor 8, common to both divisor and dividend, we have left in the dividend 2, in place of 16; canceling the factor 2, common to both divisor and dividend, we have left in the divisor 7 in place of 14; and canceling the factor 7, common to the dividend and divisor, we have left, or 3, which is the quotient.

When any factor is canceled, 1 is understood to remain, and need be written only when the last of all other factors in the dividend or divisor is canceled.

13. Divide 9 times 40 by 15 times 24.

14. Divide 75 X 25 × 7 by 50 X 3.

Ans. 1. Ans. 87.

15. (36 × 63 × 12)÷ (54 X 40 X 10)= how many? 16. (510 × 63 × 4) ÷ (680 × 84)=how many? Ans. 21. 17. Divide 400 X 189 X 33 X 5 by 320 × 126 × 11 × 5.

Ans. 5.

APPLICATIONS.

1. How many horses, worth $132 each, must be given for 1476 sheep, worth $11 each?

OPERATION.

XX X 1476

=

123, Ans.

132

12

If one sheep is worth $11, 1476 sheep must be worth $11 X 1476, and as many horses, worth each $132, must be given for $11 X 1476, as $132 are contained times in $11 X 1476, or 123.

2. How many pounds of butter, at 35 cents a pound, can be bought for 105 yards of muslin, at 21 cents a yard?

Ans. 63 pounds.

3. At 14 cents a pound, how much sugar can be bought for 2 cords of wood, at $5.60 a cord?

When a factor is canceled, what is understood to remain? When, only, need the 1 be written?

4. How many loads of hay, of 18 hundreds each, at 75 cents a hundred, will pay for 162 bushels of oats, at 50 cents a bushel? Ans. 6 loads.

5. William Marsh sold 360 pounds of beef, at 14 cents a pound, for 3 firkins of butter, each weighing 56 pounds. How much was the butter a pound? Ans. 30 cents.

to

6. How many days must a carpenter work, at $2.50 a day, pay for the services of a farmer for 40 days at $1.50 a day? 7. Sold 164 dozen school readers, at $9 a dozen, and received in payment quarto dictionaries, at $12 apiece. many dictionaries did I receive?

Ans. 123 dictionaries.

8. When $40.50 is paid for 30 barrels of apples, each containing 3 bushels, how much are they a bushel?

Ans. $.45.

9. How many bales of goods, each bale containing 60 pieces, and each piece 49 yards, worth 75 cents a yard, must be given for 80 government bonds, worth $110.25 each? Ans. 4 bales.

GREATEST COMMON DIVISOR.

120. A Common Divisor of two or more numbers is any exact divisor (Art. 97) of each of them. Thus,

2, 3, and 6 are common divisors of 6 and 12.

121. The Greatest Common Divisor of two or more numbers is the greatest exact divisor of each of them.

4 is the greatest common divisor of 8 and 12.

Thus,

But 4 is equal to the product of 2 and 2, the only common prime factors of 8 and 12 (Art. 116). Hence, the principle,

The greatest Common Divisor of two or more numbers is equal to the product of all their common prime factors.

A Factor? (98)

What is a Com

REVIEW QUESTIONS. What is an Exact Divisor? (97) A Prime Number? (99) A Composite Number? (100) mon Divisor of two or more numbers? What is the Greatest Common Divisor of two or more numbers? The Principle?

122. To find the Greatest Common Divisor of two or more Numbers.

1. What is the greatest common divisor of 8, 12, and 20?

Taking out the common prime factors of the given numbers (Art. 116), we find them to be 2 and 2; and therefore the greatest common divisor of 8, 12, and 20, is 2 X 2, or 4.

In the second operation, we resolve the given numbers into their prime factors (Art. 115), and find the common prime factors to be 2 and 2, and take their product, with the same result as before.

RULE. Find the common prime factors of the numbers, and take their product. Or,

Resolve the numbers into their prime factors, and take the product of those which are common.

Numbers prime with respect to each other (Art. 104), having no common factor, except 1, are said to have no common divisor.

The greatest common divisor of two numbers is likewise the greatest common divisor of the smaller and of the remainder after division.

8. Find the greatest common divisor of 247 and 323.

Repeat the Rule.

When are two or more numbers said to have no common divisor? What is the principle of another method?