FACTORING.

Teachers who prefer to have pupils study United States Money before Common and Decimal Fractions, are referred to p. 143.

1. What two numbers multiplied together make 6? 2. What then are the factors of 6? (P. 47, Q. 5.) 3. What are the factors of 10?

4. What are the factors of 8? 5. What are the factors of 14? 6. Name two factors of 16.

Of 12?

Of 15? Of 21?

Of 18.

Of 20.

A Factor of a number is one of the numbers, which multiplied together, produce that number. (P. 47, Q. 5.) 2. What is a Composite Number?

A Composite Number is the product of two of more factors, each of which is greater than 1. Thus, when it is said that 3 × 515, fifteen is a composite number, and 3 and 5 are its factors.

3. What is a Prime Number?

A Prime Number is one which cannot be proJuced by multiplying any two numbers together, except unit and itself.

4. What are Prime Factors?

The Prime Factors of a number are the prime numbers which, multiplied together, produce that number.

5. What is an Odd Number?

An Odd Number is one which cannot be divided by 2, without a remainder; as, 1, 3, 5, 7, etc.

6. What is an Even Number?

An Even Number is one which can be divided by 2, without a remainder; as, 2, 4, 6, 8, etc.

NOTE.-All even numbers except 2 are composite numbers.

7. What is meant by Factoring a number?

Factoring a Number is finding two or more factors which multiplied together, produce that number

MENTAL EXERCISES.

1. Name the odd numbers under 30.
2. Name the even numbers under 30.
3. Name all the composite numbers under 30.
4. Name all the prime numbers under 30.

5. What are the prime factors of 30?

ANALYSIS.-By inspection we perceive that 30 is divisible by the prime number 2, giving the factors 2 and 15. Again, dividing 15 by 3 we have the factors 3 and 5, both of which are prime Therefore, 2, 3, and 5, are the prime factors required.

6. What are the prime factors of 12? 7. What are the prime factors of 20? 8. What are the prime factors of 35?

15? 18?

28? 30?
40? 42?

SLATE EXERCISES.

1. What are the prime factors of 105?

ANALYSIS.-Dividing 105

by the prime number 3, we have the factors 3 and 35. Again, dividing the 1st quotient 35, by the prime number 5, we have the factors 5 and

OPERATION.

1st divisor, 3 105, given.
66
2d

3d

66

5 35, 1st quot.

66

7

7, 2d

I, 3d

105=3×5×7

7. Finally, dividing the 2d quotient 7 by 7. we have 7 and 1. But the divisors 3, 5, and 7 are all prime numbers, and therefore are the prime factors required.

8. How is a composite number resolved into prime factors? Divide the given number by any prime number that will divide it without a remainder. Again, divide this quotient by a prime number, and so on till the quotient is I. The several divisors are the prime factors required. NOTE. The least divisor of every number is a prime factor; hence, to avoid mistakes, it is advisable for beginners to take for the divisor, the least number that will divide the several dividends⚫ without a remainder.

Find the prime factors of the following numbers:

1. What is the quotient of 3× 3 × 5 divided by 3×5?

ANALYSIS.—3 × 3 × 5=45, and 3×5=15; now 45÷15=3. But it will be seen by inspection that the factors 3 and 5 are common to the dividend and the divisor. If we cancel or cross out the 3 in each, we have 3 × 5÷5, or 15÷5, which equals 3, the same as before. Again, if we cancel or cross out the 5 in each, we have 31, which equals 3, as before.

NOTE. To cancel a factor of a number means to erase or reject it. 2. What is the quotient of 2 × 3 × 7÷2×3×5? SOLUTION.-Cancelling the common factors 2 and 3, we have 7+5=13 Ans.

1. What is the effect of cancelling a factor from a number? It divides the number by that factor.

10. What is Cancellation?

Cancellation is the method of abbreviating operations by rejecting equal factors from the divisor and

dividend.

REMARK.—When the factor cancelled is equal to the number itself, I is always left in its place; for, dividing a number by itself, the quotient is I. When the I stands in the dividend, it mast be retained; when in the divisor, it may be disregarded.

3. What is the quotient of 2 x 5×7÷2×3×7?

5 × 1=5, that of 1 × 3 × 1=3, and 5÷3=1} Ans.

11. What is the rule for Cancellation?

Cancel all the factors common to the divisor and dividend, and divide the product of those remaining in the dividend by the product of those remaining in the divisor. 4. What is the quotient of 77 divided by 21?

SOLUTION.-By inspection, we perceive that 7 is a factor common to the divisor and the dividend. Cancelling 21, this factor, we have, or 3, Ans.

OPERATION.

7,

Perform the following divisions by cancellation.

5. 4×5×75×4×3.

6.7×3×11÷8×3×7.

7. 23×5×9÷5×7×9.

8. 28 x 13 x 11÷11 X 13× 7.

9. 63 × 39 × 2÷13×9×3. 10. 96 × 7×11÷12 × 8 × 7.

11. How many yards of cloth, at 8 dollars a yard, can be bought for 25 pair of boots, at 4 dollars a pair?

12. How many barrels of flour, at 7 dollars a barrel, must be given for 18 tons of hay, at 14 dollars a ton? 13. How long must a man work, at 3 dollars a day his rent for a year, at 11 dollars a month?

to

pay

COMMON DIVISORS.

MENTAL EXERCISES.

1. What will divide 9 and 15 without a remainder? 2. What will divide 14 and 24 without a remainder? 3. What will divide 16 and 20 without a remainder? 4. What will divide 42 and 18 without a remainder? 5. What is the greatest divisor of 18 and 27? 6. What is the greatest divisor of 12 and 36?

DEFINITIONS.

12. What is a Common Divisor?

A Common Divisor is a number which will divide two or more numbers without a remainder.

13. What is the Greatest Common Divisor?

The Greatest Common Divisor of two or more numbers, is the greatest number that will divide. each of them without a remainder.

REMARKS.-I. A common divisor of two or more numbers is always a common factor of those numbers; and the greatest common divisor of them is their greatest common factor.

2. A common divisor is often called a common measure.

3. The greatest common divisor of two or more numbers is equal to the product of all the prime factors common to those numbers.

1ST OPERATION.

16) 28 (1

16

1. What is the greatest common divisor of 16 and 28? 1ST METHOD.-Dividing the greater by the less, the quotient is 1, and 12 remainder. Again, dividing the first divisor by the first remainder 12, the quotient is 1, and 4 remainder. Next, dividing the second divisor by the second remainder 4, the quotient is 3, and o remainder. The last divisor 4, is the greatest common divisor.

12) 16 (1

12

4) 12 (3

12