100. Factoring is the method of obtaining the several factors or divisors of a number.

PRINCIPLES.

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

2. A factor of a number is also a factor of any product of that number by integers.

3. A common factor of several numbers is a factor of their sum.

4. A common factor of two numbers is a factor of their difference.

101. TO RESOLVE A NUMBER INTO ITS PRIME FACTORS. What are the prime factors of 60?

PROCESS.

2)60

ANALYSIS.-By trial, 60 divided by the prime factor 2 gives the composite factor 30 for a quotient; 30 divided by the prime factor 2 gives the composite factor 15 for a quotient; 15 divided by the prime factor 3 gives the prime factor 5 for a quotient. Hence, 2, 2, 3, and 5, are 5 the prime factors of 60.

2)30

3)15

Rule.-Divide the number by its least prime factor, and divide this and each succeeding quotient in the same manner until a quotient is obtained which is a prime number. The several divisors and the last quotient are the prime factors of the numbers.

What are the prime factors of the following numbers?

102. Cancellation is the process of facilitating operations in division by rejecting common factors from both dividend and divisor.

The Sign of Cancellation is an oblique mark drawn across the number in which is the canceled factor.

Thus, 3, 5, 6, 9.

103. The operations in cancellation are dependent upon the following

PRINCIPLES.

1. The canceling of one of the factors of a number divides that number by the factor canceled.

2. The canceling of one of a series of factors divides their product by the factor canceled.

3. The canceling of equal factors from both dividend and divisor does not change the value of the quotient.

1. Divide 3×6×8×12 by 2×3×4×6

ANALYSIS. The factors of the dividend being arranged above, and the factors of the divisor beneath a horizontal line, 3 is observed to be factor in both dividend and divisor, and is canceled from both, since by principle 3 the

value of the quotient is not changed.

In like manner, 6 is observed to be a factor of both dividend and divisor, and is canceled. 2, a factor common to 2 of the divisor and 8 of the dividend, is canceled, leaving 4 the other factor of 8, in the dividend.

4, a factor of both dividend and divisor, is canceled, leaving 12 of the dividend the only factor uncanceled, which is therefore the quotient. Hence the following

Rule.-I. Write the factors of the dividend above and the factors of the divisor beneath a horizontal line.

II. Cancel equal factors of both dividend and divisor until the factors of the one are prime to the factors of the other.

III. Divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor.

2. Divide 4×6×8×10 by 2×3×4×5. 3. Divide 5x7x9x11 by 3x5x7x11. 4. Divide 6x8x12×18 by 4×6×9×12. 5. Divide 9x12×15×18 by 6×5×9×12. 6. Divide 10×14×21 × 28 by 3×7×4×14. 7. Divide 15×20×25×27 by 10x15x18x25. 8. Divide 18×24 × 32 × 36 by 9×12×16×18. 9. Divide 20x30x35×40 by 10x15x7x8x5. 10. Divide 24×28 × 36 × 45 by 16x18x15×12. 11. Divide 21x24×28×35 by 7×14×24×5. 12. Divide 24 x 30 × 36 × 40 by 20×15×18×12. 13. (27x32× 36 × 45)÷(16x9×12×15)=? 14. (28x30x35x40)÷(14x15x20×7)=? 15. (30x35x40×45)÷(15×20×30×5)=? 16. (32 × 36 × 48×54)÷(16x18x32×27)=? 17. (36×42×45×50)÷(18×21X15X25)=? 18. (35×45×50×55)÷(15×25x35x11)=? 19. (40×48×56×60)÷(20×24×28×40) = ? 20. (45×63×72×80)÷(15×21×24×80)=?

21. A gentleman purchased 24 pounds of coffee, at 28c. a pound, and gave in payment 8 packages of sugar, each containing pounds. What is the price per pound?

22. Bought 12 car loads of coal, each containing 32 tons, worth $6 a ton, and gave in payment 8 boat loads of flour, each containing 36 barrels. What is the cost per barrel?

23. A grocer exchanged 54 firkins of butter, each containing 48 pounds, at 25c. per pound, for 12 chests of tea, each containing 60 pounds. What is the price per pound of the tea?

24. A company of wood-choppers cut 18 piles of wood of 36 cords each, and exchanged it for 12 car loads of coal of 42 tons each at $9 a ton. What is the price per cord of the wood?

25. A farmer planted 3 fields of corn of 120 rows, each row containing 72 hills, and each hill 5 grains. How many fields could he have planted with the same corn, each field having 96 rows of 75 hills each, and each hill containing 6 grains?

COMMON DIVISORS.

104. A Divisor of a number is any integer that will divide the number without a remainder.

Thus, 5 is a divisor of 10 or 15.

Since 10 and 15 are both divisible by 5, 5 is called a common divisor of 10 and 15. Hence,

105. A Common Divisor of several numbers is a divisor of each of them.

Since 5 is the greatest divisor of 10 and 15, it is called the greatest common divisor of those numbers. Hence,

106. The Greatest Common Divisor of several numbers is the greatest divisor of each of them.

ORAL.

1. What integers are divisors of 6? .8? 10? 12? 2. What integers are divisors of 9? 14? 15? 16? 3. What numbers are divisors of 18? 24? 56? 72? 4. What integers are divisors of 18 and 30? 24 and 36? 5. What numbers are divisors of 49 and 56? 63 and 72? 6. What numbers are divisors of 81 and 108? 84 and 96?

WRITTEN.

TO FIND THE GREATEST COMMON DIVISOR OF SEVERAL NUMBERS. 7. Find the greatest common divisor of 27, 36, and 45. PROCESS, BY FACTORING.

27=3×3×3

36=3x3x2x2

45=3×3×5

ANALYSIS.-Separating the numbers into their prime factors, it is found that the prime factors of 27 are 3, 3, and 3; the prime factors of 36 are 3, 3, 2, and 2; and

the prime factors of 45 are 3, 3, and 5. Upon inspection it appears that the only prime factors found in each of these numbers are 3 and 3. Therefore, 3×3=9 is the greatest divisor of each of the numbers, and is therefore their greatest common divisor. Hence the following Rule. Resolve the numbers into their prime factors, and find the product of the factors which are common.

PRINCIPLES.

1. A divisor of a number is a divisor of any product of that number by integers.

2. The greatest common divisor of several numbers is either the least of the numbers or a factor of the least.

Find the greatest common divisor of the following numbers: 8. 24, 48, 64. 24. 280, 350, 420.