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76. The pupil may be required to illustrate the following problems by original examples:

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2. The greater and the less of

two numbers, or the minu-the difference or remainder. end and subtrahend,

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eral factors, and the prod- that one factor.
uct of all but one factor,

FACTORS AND DIVISORS.

INDUCTIVE EXERCISES.

7. 1. What two numbers, besides the number itself and 1, will give a product of 8? 16? 25? 42? 64?

2. What numbers, other than the given number and 1, will exactly divide 9? 15? 36? 48? 55?

3. Of what sets of two numbers is 24 the product?

4. Of what sets of three numbers is 36 the product?

5. What are the smallest numbers, other than 1, that will exactly divide 18? 21? 49? 55?

6. What is the largest number, other than the given number itself, that will exactly divide 22? 24? 30? 40? and 30, that are the Between 30 and 50.

7. Name the numbers between 12 product of two factors greater than 1. 8. Name the numbers between 5 and 20, that have no other factors than the numbers themselves and 1.

DEFINITIONS.

78. The integral factors of a number are the integers whose continued product equals that number.

Thus, 4 and 5 are factors of 20; 2, 4, and 8, of 64.

79. A composite number is a number that has other integral factors besides itself and 1.

Thus, 24 is a composite number, since it is the product of 4 and 6, or 3 and 8, or 2, 3, and 4.

80. A prime number is a number that has no integral factors besides itself and 1.

Thus, 2, 3, 5, 7, 11, etc., are prime numbers.

81. A prime factor is a prime number used as a factor. Numbers are said to be prime to each other when they have no common integral factors or divisors.

Thus, 9 and 4, 12 and 25 are prime to each other.

82. An exact divisor of a number is any integral factor of that number.

Thus, 3 and 4 are exact factors or divisors of 12.

1. An exact divisor of a number is also called a measure of that number. 2. When any integral factor is common to two or more numbers, it is called a common divisor, factor, or measure of those numbers.

83. An even number is one that is exactly divisible by 2. All numbers terminating with 0, 2, 4, 6, or 8 are even.

84. An odd number is one that is not exactly divisible by 2.

All numbers terminating with 1, 3, 5, 7, or 9 are odd.

Any number is exactly divisible

1. By 2, when it is an even number.

2. By 3, when the sum of its digits is divisible by 3.

3. By 4, when its units and tens together are divisible by 4. 4. By 5, when it ends with a 0 or 5.

5. By 6, when it is an even number and divisible by 3.

6. By 8, when its three right-hand figures are ciphers, or express a number divisible by 8.

7. By 9, when the sum of its digits is divisible by 9.

8. By 10, when it ends with one or more ciphers.

Find by inspection some of the exact divisors of the follow

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85. 1. What are the prime numbers from 8 to 24? 2. What are the composite numbers from 8 to 24 ? 3. Name all the prime factors of 36.

4. Name all the composite factors of 36.

5. What prime factors are common to 21 and 42 ?

6. What composite factors are common to 36 and 72?

7. What factors are common to 16 and 24? To their sum? To their difference?

8. Form a composite number by using 2 twice as a factor; 3 twice as a factor; 2 three times as a factor.

9. What is one of two equal factors of 9? of 16? of 25? 10. One of three equal factors of 8? of 27? of 64?

When a composite number is composed of equal factors, it is called a power of that factor; and one of the equal factors is called a root of the number.

The power of a number is named according to the number of equal factors used. Thus,

3x3 is the second power or square of 3.

3x3x3 is the third power or cube of 3. 3x3x3x3 is the fourth power of 3.

So, also, 3 is called the second root or square root of 9; the third root or cube root of 27.

11. Required the second power of 3; of 4; of 5; of 6; of 8; of 9; the third power of 2; of 3; of 4; of 5; of 6.

12. What is one of two equal factors of 9? of 25? of 36? 13. One of three equal factors of 8? 27? 64? 125? 14. What is the second root of 9 ? of 16? of 25? of 64? 15. What is the third root of 8 ? of 27? of 64? of 125?

Instead of repeating the factor, it is generally written but once, and a small figure, called the exponent, is placed at the right, and a little above the numbers, to show how many times it is used as a factor. Thus,

34 = 3x3x3x3, and denotes that 3 is used as a factor 4 times.

63 = 6 × 6 × 6, and denotes that 6 is used as a factor 3 times.

PRINCIPLES.-I. Every composite number is the product of all its prime factors.

II. A factor common to two or more numbers is a factor of their SUM; also, of the DIFFERENCE of any two of them.

86. Factoring is the process of separating a composite number into its integral factors.

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8%. To resolve a number into its prime factors.

1. What are the prime factors of 252 ?

ANALYSIS.-Since the given number is even, divide by the least prime factor 2, and the quotient also by 2.

Next divide by the prime numbers 3 and 3 successively, obtaining 7 for the last quotient, which being prime, the division can be carried no further. Hence the divisors 2, 2, 3, 3, and the last quotient 7 are all the prime factors or divisors of 252, and may be written 7, 32, 22. PROOF.-7×3×3×2×2 = 252.

In like manner, find the prime factors or divisors

2|252

2126

3

63

3

21

7

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The principles and the operation make the rule obvious.

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