FACTORS.

92. The Factors of a number are the integers which being multiplied together produce the number.

Thus, 6 and 3 are the factors of 18. 5, 2, and 3 are the factors of 30.

1. Name one of the factors of 16. Name three factors of 16.

2. What are the factors of 24, 20, 36, 48, 60, 21, 44, 56, 63, 50, 81, 64?

3. What numbers will exactly divide 54, 48, 63, 24, 35, 84, 77, 96?

93. A number that has other factors besides itself and 1 is a Composite Number. 6, 8, 15, are composite numbers. A composite number is divisible by any of its factors. Its factors are, therefore, exact divisors of it.

4. Name three composite numbers and their factors. Name three numbers that are not composite.

94. A number that has no factors except itself and 1 is a Prime Number. 5, 7, 11, 17, are prime numbers.

5. Name three prime numbers. Name all the prime numbers from 1 to 30,

95. A prime number used as a factor is a Prime Factor. 6. 3 and 5 are prime factors of 15, but 4 and 6 are not prime factors of 24. Why not?

NOTE. - Every number may be multiplied by 1 to produce itself. While both are factors, according to the definition, they are not so used in practice.

7. Name the prime factors of 20, 18, 24, 25, 27, 38, 34, 48, 49, 63.

96. A number that is divisible by 2 is an Even Number. Thus, 8, 6, 10, 14, are even numbers.

8. Name all the even numbers from 1 to 40.

97. A number not divisible by 2 is an Odd Number. Thus, 3, 5, 9, 11, are odd numbers.

9. Name the odd numbers from 1 to 40.

98. Every prime number except 2 and 5 ends with 1, 3, 7, or 9.

10. Name a prime number ending with 1, with 3, with 7, with 9.

11. Name a composite number ending with 1, with 3, with 7, with 9.

2 is an exact divisor of any even number, or of any number ending with an even number, or with 0.

3 is an exact divisor of any number the sum of whose digits is divisible by 3. 4 is an exact divisor of any number when the number expressed by its two right-hand figures is divisible by 4.

5 is an exact divisor of any number ending with 0 or 5. Separating a number into its factors is called Factoring. 99. Written.

12. Find the prime factors of 60.

2/60

2 30 3/15

55

Rule.

1

SOLUTION. We first divide by the prime number 2. The quotient, 30, being even, we also divide by 2. The quotient is 15, which we divide by 3, giving a quotient of 5, which we divide by 5. The last quotient is 1. Hence the prime factors of 60 are 2, 2, 3, 5.

- Divide the given number by any prime number that will exactly divide it. Divide this quotient by any prime number, and so continue until the quotient is 1. The several divisors are the prime factors.

33. What is the quotient of 42 ÷ 21?

34. What is the quotient of 3 × 2 × 7 divided by 3 × 7? 35. How many times is 3 times 5 contained in 6 times 5? 3 times 8 in 12 times 8?

36. Divide 12 × 16 by 4 × 16. 9 × 11 by 3 × 11.

37. Divide 18 x 7 by 9 × 7.

38. In the last example what factor is found in both dividend and divisor?

39. Would the quotient be the same if the factor 7 were rejected from both dividend and divisor?

100. PRINCIPLES. -1. Rejecting a factor from a number divides the number by that factor.

2. Rejecting the same factors from both dividend and divisor does not affect the quotient. (See Principles of Division.)

101. Rejecting the same factors from both dividend and divisor is called Cancellation.

2

102. 1. Divide 8 × 18 × 15 × 7 by 4 × 6 × 11 × 9.

SOLUTION. - We first indicate the division by writing the dividend

5 8 × 18 × 13 × 7 4 × 6 × 11 × 9

3

over the divisor with a line between. Since 4 and 6 are factors of 8 and 18, respectively, they may be omitted, or cancelled, from both dividend and divisor. Since 3 in the dividend is a factor of 9 in the divisor, it is cancelled from

both, leaving 3 in the divisor. 3 in the divisor, being a factor of 15 in the dividend, is cancelled from both.

The product of the uncancelled factors in the dividend is 70, and in the divisor 11. The quotient is therefore, which equals 6. Indicate, and find quotients by cancellation.

2. Divide 36 x 27 x 49 x 38 x 50 by 70 x 18 x 15.

3. (28 x 38 x 48) + (14 x 19 x 24 x 2 x 2)=?

4. (26 × 5 × 54) ÷ (13 × 5 × 6) = ?

5. What is the quotient of 36 × 48 × 16 divided by 27 x 24 x 8?

6. Divide 5 × 45 × 7 × 20 by 49 × 5 × 4 × 9.

7. Divide 5 × 51 × 7 × 9 × 4 by 17 × 20 × 12 × 7 × 2.

8. Divide 25 × 2 × 72 × 14 by 6 × 9 × 120.

9. How many bushels of potatoes at 50 cents a bushel must be given in exchange for 15 pounds of tea at 40 cents a pound?

10. If 60 yards of cloth cost $120, how many yards can be bought for $40?

11. 15 oranges cost 45 cents. cost?

How much will 7 oranges

12. A dairyman sells 100 quarts of milk daily at 5 cents a quart. How many bushels of corn at 45 cents a bushel can he buy with 10 days' milk receipts?

13. A farmer sold a grocer 45 bushels of apples at 50 cents a bushel, taking his pay in flour at 90 cents a sack. How many sacks did he receive?

14. In what time can a boy, at 60 cents a day, earn as much as a man can earn in 40 days at $3 a day.

NOTE. Change the $3 to cents.

15. If 32 quarts of chestnuts cost $2.50, what will 800 quarts cost?

16. There are 16 ounces in a pound. 30 pounds of steel will make how many horseshoes, each weighing 6 ounces?

17. A man sold 15 cords of wood at $6 a cord and received payment in wheat at 90 cents a bushel. How many bushels of wheat did he receive?

18. If I buy 10 yards of cloth at $ 2 a yard, and pay for it in wool at 50 cents a pound, how many pounds of wool will it require?

19. If wood is worth $4 a cord, and coal $5 a ton, how many cords of wood will pay for 20 tons of coal?

20. Divide the product of 18, 6, 9, and 4 by the product of 10, 7, 6, and 2.

21. Divide the product of 10, 75, 9, and 96 by the product of 5, 12, 15, and 9.

22. Find the quotient of 51 times the product of 54 and 12 divided by 36 times the product of 17 and 3.

23. How many jars of butter each containing 10 pounds at 20 cents a pound must be given for 10 sacks of granulated sugar each containing 5 pounds at 5 cents a pound?

24. A grocer sold 20 boxes of soap, each containing 100 packages at 4 cents a package, and took in payment hay at $16 a ton. How many tons did he receive?

GREATEST COMMON DIVISOR.

103. 1. What number will exactly divide 12 and 15? 12 and 36? 15 and 20?

NOTE.

When two or more numbers have the same factor, it is

called a common factor of those numbers.

2. Name a common factor of 9, 12, and 15.

3. What factor is common to 15, 20, and 25? What factor is common to 14, 21, and 28?

4. Name two common divisors of 10 and 20.

5. Name the greatest factor that is common to both 18 and 30.