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 the 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 or by some product of them, or by unity. OPERATION. CASE 1. 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.-1. Divide the given number by any prime number greater than 1, that will exactly divide it. II. Divide the quotient, if composite, in the same manner, und thus continue until the quotient is preme. III. The divisors and last quotient unll be the prime factor: required. What are the prime factors 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, 3, 7, 7, 13. 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 5)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.-1. 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, 196, 256, 324, 900, 1296, 2025. Ans. 4, 4; 6, 6; 10, 10; 14, 14, etc. 3. Find the square root of 225, 576, 1764, 3136, 3969.5184 Ans. 15; 24; 42 ; 56; 63; 72. 4. Find the cube root of 27, 64, 125, 216, 512, 1000. Ans. 3; 4; 5; 6; 8; 10. 5. Find the fourth root of 16, 81, 256, 1296, 4096, 20736 Ans. 2; 3; 4; 6; 8; 12. 6. 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 15? to 19 and 18? to 24 and 36? to 32 and 48? to 48 and 72 ? 3. What may a divisor common to two or more numbers be callout Ans. Their common divisor. 4. What is a common divisor of 16 and 247 of 15 and 207 of 18 and 80? of 48 and 54 ? 5. What is the greatest divisor common to 24 and 32? to 32 and 567 to 48 and 72? to 72 and 96 ? 6. What may the greatest divisor common to two or more numbers be cailed ? Ans. Their greatest common divisor. 7. What is the greatest common divisor of 24 and 307 of 45 and 50? uf 54 and 60 ? of 64 and 72? 8. What prime factors are common to 18 and 24? 27 and 307 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 that 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 divisor may be represented by the Initiale G. C. D. PRINCIPLES. 1. A common factor of two or more numbers is a factor of 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. 42 120. When the numbers are small and can be readWy factored. FIRST METHOD. 121. This method consists 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 321 a factor of the G. C. D. (Prin. 1). Dividing the 63 quotients by 3, we see that 3 is a factor of each 7 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 X3x7=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. III. Take the product of all the divisors; the result wili be the greatest common divisor. What is the greatest common divisor of 2. 10, 20, and 30? Ans. 10. 3. 36, 48, and 54? Ans. 6. 4. 18, 36, and 72? Ans. 18. 5. 48, 72, and 96 ? Ans. 24. 6. 120, 210, and 360 ? Ans. 30. 7. 210, 315, and 420 ? Ans. 105. 8. 252, 336, and 420? Ans. 84. 9. 330, 495, and 660? Ans. 165. 10. 468, 780, and 1092? Ans. 156. SECOND METHOD. OPERATION. 122. This method consists in resolving the numbers into their prime factors, and taking the product of the com. mon factors. 1. Find the greatest common divisor of 42, 84, and 126. SOLUTION.—The factors of 42 are 2, 3, and 7; the factors of 84 are 2, 2, 3, and 7; the factors of 42=2X3X7 126 are 2, 3, 3, and 7. We see that 2, 3, and 7 8452X2X3X7 are all the prime factors common to the three 126=2X3X3X7 numbers; hence their product, which is 42, is the 2x3x7=42 greatest common divisor of the numbers (Prin. 2). Hence the following Rule.-Resolve the numbers into their prime factors, and take the product of all the common factors. Find the greatest common divisor of 2. 270, 315, and 405. Ans. 45. 3. 168, 192, and 216. Ans. 24. 4. 252, 308, and 364. Ans 28. 6. 504, 546, and 588. Ans 42. 6. 392, 448, and 504. Ans. 56. 7. 432, 504, and 648. Ans. 72. 8. 792, 864, 936, and 1008. Ans. 72. 9. 384, 576, 768, and 960. Ans, 192. OPERATION. CASE II. 123. When the numbers are large and cannot be readily factored. 1. Find the greatest common divisor of 32 and 56. SOLUTION.–We divide 56 by 32, the divisor 32 by 24, and the divisor 24 by the remainder 8, and 32)56(1 have no remainder; then will 8 be the greatest 32 common divisor of 32 and 56. For 1st. The last remainder, 8, is a NUMBER OF TIMES 24)32(1 the G. C. D. Since 32 and 56 are each a number of 24 times the G. C. D., their difference, 24, is a number 8)24(3 of times the G. C. D., by Prin. 3; and since 24 and 24 32 are each a number of times the G. C. D., their difference, 8, is also a number of times the G. C. D. 2d. The last divisor, 8, is ONCE the G. C. D. Since 8 divides 24 it will divide 24+8, or 32, by Prin. 3; and since it divides 32 and 24, it will divide 32+ 24, or 56; and now since 8 divides 32 and 56, and is a number of times the G. C. D., it must be once the G. C. D. |