14. I sold 5 dozen of eggs at the rate of 4 for 7 cents, and received 5 cents in money, and the balance in coffee at 25 cents a pound. How many pounds of coffee did I receive?

94.-Ex. 1. If in making mortar, 6 bushels of sand are required for each cask of lime used, how many bushels of sand will be required for 97 casks of lime? Ans. 582.

2. How many loads, each containing 1345 bricks, are there in a pile containing 286485 bricks?

3. If the front and rear walls of a house each require for their construction 31650 bricks, and the other two walls each require 43400 bricks, how many bricks are required for the four walls? Ans. 150100.

4. The factors of a product are 3043 and 405. What is that product? Ans. 1232415. 5. The product of two factors is 9225; one of the factors is 45. What is the other factor?

45)9225(205

90

225

225

SOLUTION. Since a product is the result obtained by multiplying one of two factors by the other, the quotient obtained by dividing the product by one of the factors must be the other factor. 9225 45 205, the factor required.

6. If a product is 9225 and the multiplicand 205, what is the multiplier?

7. If a man has 15000 dollars, and should spend enough of it to pay for a farm of 80 acres at 78 dollars per acre, how much would he have left?

8. If a merchant should purchase 1011 barrels of flour at 12 dollars a barrel, and pay down 7919 dollars, how much would he then owe for the flour? Ans. 4213 dollars.

What is the

Ans. 630.

9. The dividend is 15750 and the divisor 25. quotient, or the other factor of the dividend? 10. What is the average of 16, 22 and 28? SOLUTION.-The average of two numbers is one half of the sum of those numbers; the average of three numbers is one third of their sum, etc. The sum of 16, 22 and 28 is 66, and one third of 66 is 22.

11. The elevation of the northern lakes above the sea is as follows: Superior, 627 feet; Michigan, 587 feet; Huron, 574 feet; and Ontario, 282 feet. What is their average elevation above the sea? Ans. 5174 feet. 12. What is the value of (14+6) + 16 × 2 (63 194) 459-3?

(14+6) + 16 × 2 − (63—19 +4)+45÷9—3 =20+ 16 X 240+45÷9-3

=20+32-40+5-3; = 57-43; = 14

SOLUTION. Combining first the numbers in the parentheses, we have 20 for the value of (14+6), and 40 for the value of (63 — 19 + 4). Combining the numbers affected by the signs of multiplication and those by the sign of division, we have 32 for the value of 16 X 2, and 5 for 459. Then, combining the numbers as indicated by the signs of addition and subtraction, we have 14 as the value required.

13. What is the value of 4 + 6 × 5-16÷8-(4 X 2)? 14. Henry has 7375 dollars, which is 7 times as much as Daniel has, lacking 780 dollars. How many dollars has Daniel? Ans. 1165.

15. Two candidates at an election received in the aggregate 15653 votes. If one of them received 783 votes more than the other, how many votes did the other receive?

SECTION IX.

FACTORS AND DIVISORS.

95.-Ex. 1. Of what two integers is 6 the product? 2. What two integers multiplied together produce 15? What two produce 21?

3. What integers are factors of 6? Of 15? Of 21? 4. Of what three integers is 30 the continued product? 5. Of what sets of two integers is 30 the product?

30=2X 15, or 3 X 10, or 5 X 6.

6. Of what sets of integers greater than 1 is 24 the product? 7. Name some numbers that are the product of integers greater than 1.

8. What are the smallest integers greater than 1 that will divide 21 without a remainder? 30 without a remainder?

9. Give the sets of integers greater than 1 which, when multiplied together, will produce 30.

10. Of what number are 3 and 5 the factors? 2, 3 and 5 the factors?

DEFINITIONS.

96. The Factors of a number are the integers which being multiplied together will produce that number.

Thus, 2, 3 and 5 are the factors of 30; for 2 × 3 × 5 = 30.

97. A Prime Number is an integer that has no factor except itself and 1.

Thus, 1, 3, 5, 7 and 11 are prime numbers.

98. A Composite Number is an integer that has other factors besides itself and 1.

Thus, 4 and 6 are composite numbers, since 4-2X2; and 6 = 2 × 3. 99. A Prime Factor is a factor which is a prime number. The prime factor 1 is not commonly mentioned, since it is a factor of every integer.

Numbers are said to be mutually prime, or prime to each other, when they have no common factor except 1.

100. Factoring is the process of finding the factors of composite numbers.

The number of times a number is taken as a factor may be denoted by writing a small figure, called an Exponent, at the right and above the figure or figures of the factor.

Thus, 323 X 3, and denotes that 3 is taken twice as a factor.

113 = 11 X 11 X 11, and denotes that 11 is taken 3 times as a factor.

101. An Exact Divisor of a number is any integer which will divide the number without a remainder.

Thus, 1, 2, 3, 4, 6 and 12 are each exact divisors of 12.

The Exact Divisors of a number are called, also, Divisors, or Measures, of that number, and must be factors of it.

A number is said to be divisible by its exact divisors.

Thus, 12 is divisible by its exact divisors 1, 2, 3, 4, 6 and 12.

102. Any number is divisible by 2 when its right-hand figure is 0, 2, 4, 6, or 8.

For such numbers are composed of some exact number of twos.

Numbers divisible by 2 are Even Numbers, and all others are Odd Numbers.

103. A number is divisible by 4 if its tens and ones are divisible by 4.

For 4 is an exact divisor of 100, and of any number of hundreds; hence, if the tens and ones of a number are divisible by 4, the number itself must be.

Thus, 648 and 7312 are each divisible by 4.

104. A number is divisible by 5 if its right-hand figure is 0 or 5.

A number whose right-hand figure is 0 is an exact number of tens; a number whose right-hand figure is 5 is an exact number of tens plus 5. 5 is an exact divisor of 5 or 10; hence, any exact number of tens, or any exact number of tens plus 5, is divisible by 5.

Thus, 70 and 75 are each divisible by 5.

105. A number is divisible by 3 or 9 when the sum of the ones represented by its figures is divisible by 3 or 9.

Take, for example, the number 7542; 7542=7000+500 + 40+ 2; and 70007 times 999 +7; 5005 times 99+5; 404 times 9+4; 2; where the figures expressing the number of each order plus

and 2

=

the exact number of times 9 are the figures of the given number. Now, 7 times 999, 5 times 99 and 4 times 9, being each divisible by 9 and by 3, if the sum of the ones represented by the figures of the number are divisible by 9 or by 3, the number itself is thus divisible.

Thus, 7542 and 9765 are each divisible by 9 and by 3.

106. Principles.—1. Every number is equal to the product of all its prime factors.

2. A number is divisible by all its prime factors, and by ali the products of two or more of them, and is divisible by no other numbers.

WRITTEN EXERCISES.

107.-Ex. 1. What are the prime factors of 70?

2)70
5)35

7

Proof, 2X5X7=70

SOLUTION. Since the right-hand figure is 0, we can divide by the prime numbers 2 and 5. (Arts. 102 and 104.)

Dividing by these prime numbers gives for a quotient 7, which is also prime. Hence, the prime factors of 70 are 2, 5 and 7.

2. What are all the factors or divisors of 66?

2)66

3)33

11

2, 3, 11 3X2=6 11X2=22

11 X 3=33

11 X3 X 2=66

SOLUTION. Since every prime factor of a number, and every product of two or more of these prime factors, is an exact divisor of the number, and no other numbers can be exact divisors of

that number (Art. 106—2), the prime factors 2, 3 and 11, and the products of 3 by 2, 11 by 2, 11 by 3, and 11 by 2 times 3, must be all the factors or exact divisors of 66.

Hence, 2, 3, 11, 6, 22, 33 and 66 are the factors and divisors required.

3. What are the prime factors of 84? Ans. 22, 3 and 7. 4. What are all the factors or divisors of 56?

Ans. 23, 7, 4, 8, 14, 28, 56.

108. Rule for Factoring.-Divide the given number by any of its prime factors greater than 1. Divide the quotient, if composite, in like manner, and so proceed