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If the three right hand terms are ciphers, the number eqnals a number of thousands, and since 1000 is divisible by 8, any number of thou. sands is divisible by 8.

If the number expressed by the three right hand digits is divisible by 8, the entire number will consist of a number of thousands plus the num. ber expressed by the three right hand digits (thus 17368517000 +368) and since both of these parts are divisible by 8, their sum, which is tho number itself, is divisible by

7. A number is divisible by 9 when the sum of the digits is divisible by 9.

This may be shown by trying several numbers, and, seeing that it is true with these, we can infer that it is true with all. It may also be rigidly demonstrated.

8. A number is divisible by 10 when the unit figure is 0. For, such a number equals a number of tens, and any number of tens is divisible by 10, hence the number is divisible by 10.

NOTE.-1. A number is divisible by 7 when the sum of the odd numerical periods, minus the sum of the even numerical periods, is divisible by 7.

2. A number is divisible by 11 when the difference between the sums of the digits in the odd places and in the even places is divisible by 11, or when this difference is 0.

3 These two principles are rather curious than useful. For their do monstration see Higher Arithmetic.

INTRODUCTION TO FACTORING.

MENTAL EXERCISES.

1. Name the prime numbers from 1 to 50.
2. Name the composite numbers from 0 to 50.
3. Name some of the factors or makers of 12, 15, 21, 28, 36, 54, 72.

4. Name the prime numbers which are factors of 12, 18, 20, 24, 36, 54, and 72.

5. What shall we call the factors of numbers when they are primo numbers? Ans. The Prime Factors.

6. Name the prime factors of 12, 16, 18, 20, 24, 30, 32, 36, 40, 45, 50, 60, and 80.

7. Illustrate the principle that the factors of a number are divisors of the number.

8. How then can we find the factors of a number? Ans. By and ing the divisors of a number.

9. How can we find the divisors of a number? Ans. By trial, aided by the principles of Art. 110.

10. What do we call that subject of arithmetic which treats of find. ing the factors of num pers? Ans. Factoring.

11. How then may we define the subject of Factoring?

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FACTORING. 111. Factoring is the process of finding the factors of composite numbers. Unity and the number itself are not regarded as factors.

112. The Factors of a composite number are the numbers which multiplied together will produce it; thus, 3 and 4 are factors of 12.

113. The Prime Factors of a composite number are the prime numbers which multiplied together will produce it; thus, 2, 2, and 3 are the prime factors of 12.

114. One of the two equal factors of a number is called its 2d, or square root one of the 3 equal factors, its 3d, or cube root, etc.; thus, 3 is the 2d root of 9, 2 tbe 3d root of 8.

PRINCIPLES. 1. A divisor of a number, excepting unity and itself, is a factor of that number.

2. A divisor of a factor of a number, excepting unity, is a factor of the number.

3. A number is divisible by its prime factors or by any product of them.

4. A number is divisible only by its prime factors sing some product of them, or by unity.

CASE 1.

OPERATION.

115. To resolve a number into its prime factors. 1. Find the prime factors of 105. SOLUTION.-Dividing by 3, we find that 3 is a factor of 105 (Prin. 1). Dividing the quotient by 5, we 3)105 find that 5 and 7 are factors of 35 (Prin. 2), and since these numbers 3, 5, and 7, are prime numbers, they are

5)35 be prime factors of 105.

7 Rule.-I. Divide the given number by any prime number greater than 1, that will exactly divide it.

II. Divuile the quotient, if composite, in the same manner, and thus continue until the quotient is prime.

III. The divisors and last quotient wil be the prime fac tors required.

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What are the prime factors 2. Of 287 3. Of 84? 4. Of 125 ? 5. Of 325 ? 6. Of 210? 7. Of 114? 8. Of 4321 9. Of 426 ? 10. Of 1872 ? 11. Of 7644 ? 12. Of 1184 ? 13. Of 1140 ?

Ans. 2, 2, 7. Ans. 2, 2, 3, 7. Ans. 5, 5,

5 Ans. 5, 5, 13. Ans. 2, 3, 5, 7. Ans. 2, 3,

19. Ans. 2, 2, 2, etc. Ans. 2, 3,

71 Ans. 2, 2, 2, 2, 3, 3, 13.

Ans. 2, 2, 3, 7, 7, 13.
Ans. 2, 2, 2, 2, 2, 37.

Ans. 2, 2, 3, 5, 19.

CASE II.

116. To resolve a number into equal factors. 1. Find the two equal factors of 1225.

OPERATION.

SOLUTION.- We first resolve the number into its

6)1225 prime factors. Now since there are two 5's, we take one 5 for each factor; and since there are two 7's, we

5)245 take one 7 for each factor; hence each of the two 7)49 equal factors is 5x7, or 35. Therefore 35 is one of

7 the two equal factors of 1225.

5x7=35 Rule.-I. Resolve the number into its prime factors.

II. Take the continued product, of one of each of the two equal factors, when we wish the two equal factors, one of each of the three, for the three equal factors, etc.

2. Find the two equal factors of 16, 36, 100, 190, 256, 324, 900, 1296, 2025. Ans. 4, 4; 6, 6; 10, 10; 14, 14, etc. 3. r'ind the square root of 225, 576, 1764, 3136, 3969, 5184

Ans. lu; 24; 42 ; 56; 63; 72. 4. Find tbe cube root of 27, 64, 125, 216, 512, 1000.

Ans. 3; 4; 5; 6; 8; 10. 6. Find the fourth root of 16, 81, 256, 1296, 4096, 20736

Ans. 2; 3; 4; 6; 8; 12. 8. Find the fifth root of 32; 243; 1024; 3125.

Ans. 2; 3; 4; 5

INTRODUCTION TO COMMON DIVISOR.

MENTAL EXERCISES, 1. Name an exact divisor of 6; of 9; of 12; of 15; of 18; of 20; of 84; of 36.

2. What exact divisors are common to 8 and 12 ? to 10 and 157 to 12 and 187 to 24 and 367 to 32 and 48? to 48 and 72 ?

3. What may a divisor common to two or more numbers be callod? Ans. Their common diorsor.

4. What is a common divisor of 16 and 247 of 15 and 207 of 18 and 807 of 48 and 54 ?

5. What is the greatest divisor common to 24 and 327 to 32 and 569 to 48 and 729 to 72 and 96 ?

6. What may the greatest divisor common to two or more numbers be called ? Ans. Their greatest common divisor.

7. What is the greatest common divisor of 24 and 307 of 45 and 50? of 54 and 607 of 64 and 72 ?

8. What prime factors are common to 18 and 24? 27 and 80? 30 and 35? 36 and 40 ?

9. The product of what two prime factors of 12 and 18 will divide both? of 20 and 30 ?

GREATEST COMMON DIVISOR. 117. A Divisor of a number is a number tbat exactly divides it. Thus 4 is a divisor of 20.

118. A Common Divisor of two or more numbers is a number that exactly divides each of them. Thus 4 is a common divisor of 16 and 20.

119. The Greatest Common Divisor of two or more numbers is the greatest number that exactly divides each of them. Thus 8 is the greatest common divisor of 16 and 24.

NOTE.—The greatest common wivisor may be represented by the initials G. C. D.

PRINCIPLES.

1. A common factor of two or more numbers is a factor off their greatest common divisor.

2. The product of all the common prime factors of two or more numbers is their greatest common divisor.

3. A common divisor of two numbers is a divisor of their sum and also of their difference.

DEM.—Take any two numbers, as 12 and 20, of which 4 is a common divisor. Now, 12 equals three times 4, and 20 equals five times 4, and their sum equals three times 4 plus five times 4, or eight times 4, which contains 4, or is divisible by 4. Their difference is five times 4 minus three times 4, or two times 4, which is also divisible by 4.

CASE I.

OPERATION.

120. When the numbers are small and can be readUy factored.

FIRST METHOD. 121. This method consiowd in finding the common factors, and taking their product. (1. Find the greatest common divisor of 42, 84, and 126.

SOLUTION.-We write the numbers one beside another, as in the margin. Dividing by 2, we see 2/42 - 84 - 126 that 2 is a factor of each number; it is therefore a factor of the G. C. D. (Prin. 1). Dividing the

321 42 63 quotients by 3, we see that 3 is a factor of each 71 7 - 14 - 21 number and therefore a factor of the G. C. D.;

1 2 3 and in the same way we see that 7 is a factor of the G. C. D. Now since the quotients 1, 2, and

2 X 3 X 7=42 3 are prime to each other, 2, 3, and 7 are all the common factors; hence their product, which is 42, is the G. C. D. (Prin. 2).

Rule.--I. Write the numbers one beside another, with a vertical line at the left, and divide by any common factor of all the numbers.

II. Divide the quotients in the same manner, and thus continue until the quotients have no common factor.

LII. Take the product of all the divisors; the result will

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Loe the greatest common divisor.

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What is the greatest common divisor of
2. 10, 20, and 30?
3. 36, 48, and 54 ?
4. 18, 36, and 72?
6. 48, 72, and 96 ?
6. 120, 210, and 360 ?
7. 210, 315, and 420?
8. 252, 336, and 420 ?
9. 330, 495, and 660 ?
10. 468, 780, and 1092 ?

Ans. 10.

Ans. 6. Ans. 18. Ans. 24. Ans. 30. Ans. 105.

Ans. 84. Ans. 165. Ans. 156

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