INTRODUCTION TO SECONDARY OPERATIONS. MENTAL EXERCISES. 1. What numbers multiplied together will produce 4? 6? 8? 10! 12? 14? 16? 18? 20? 24? 26 ? 28 ? 30 ? 2. What numbers can be produced out of the numbers 2 and 3? 3 and 5 ? 2, 3, and 5? 3, 4, and 5? 2, 3, 4, and 5? 3. Will the product of any two numbers, each greater than a unit produce 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37 ? 4. What may we call a number which is composed by multiplying several numbers together? Ans. A Composite Number. 5. What shall we call numbers that cannot be produced by multiplying several numbers together? Ans. Prime Numbers. 6. Which are prime and which composite numbers in the follow ing list : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15? 7. What may we call the numbers whose product makes a composite number? Ans. Makers of the numbers. 8. If the word Factor means the same as maker, what may we call the makers of a composite number? Ans. Factors. 9. Form composite numbers out of the factors 3 and 4; 3, 4, and 6; 4, 5, and 6. What are the factors of 12? 15? 18? 20? 21? 24? 10. Form a composite number by using 2 twice as a factor; 3 twice as a factor; 2 three times as a factor; 3 four times as a factor. 11. Required one of the troo equal factors of 9; of 16; of 25; of 36. one of the three equal factors of 8; of 27; of 64; of 125. 12. A number composed of two equal factors is called the second power of that factor; of three equal factors, the third power, etc. 13. Required the second power of 3; of 4; of 6; of 7; of 8: the third power of 2; of 3; of 4; of 6. 14. One of the two equal factors of a number is the second root of 8 numl er; one of the three equal factors is the third root, etc. 15. What is the second root of 16? of 25 ? of 36? of 49? What is the third root of 87 of 27? of 64? of 125 ? 16. What would it seem natural to call the process of making composite numbers ? Ans. Composition. 17. What would it seem natural to call the process of finding the factors of a number? Ans. Factoring. 18. What are the first four operations of arithmetic called ? Ans. The Fundamental or Primary Operations of arithmetic. 19. What would it be natural to call these operations which are derived from the fundamental operations? Ans The Derivative or Secondary Operations. SECTION III. SECONDARY OPERATIONS. 100. The Primary Operations of arithmetic are those of synthesis and analysis, including the four fundamental rules. 101. The Secondary, or Derivative Operations, are those which arise from or grow out of the primary operations of synthesis and analysis. 102. The Secondary Operations are Composition, Factoring, Greatest Common Divisor, Least Common Mula tiple, Involution, and Evolution. COMPOSITION. 103. Composition is tho process of forming composite numbers when their factors are given. 104. A Composite Number is a number which can be produced by multiplying together two or more numbers, each greater than a unit; as 8, 12, 15, etc. 105. The Factors of a composite number are the numbers, which, when multiplied together, will produce it; thus 4 and 2 are the factors of 8. 106. A Prime Number is one that cannot be produced by multiplying together two or more numbers, each greater than a unit; as, 2, 5, 7, 11, etc. 107. A composite number consisting of two equal faciors is said to be the 2d power of that factor; of three equal factors, the 3d power, etc.; thus, 9 is the 2d power of 3, and 64 is the 3d power of 4. NOTE.-An even number is one that is exactly divisible by 2; an odd Qumber is one that is not exactly divisible by 2. MENTAL AND WRITTEN EXERCISES. 1. Tell which of the following numbers are prime or composite : 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. 2. Name the prime numbers from 1 to 53. Name the prime numbers from 53 to 101. 3. Write the numbers from 1 to 100, and cut out all tbe composite numbers, leaving the primes. 4. What is the second power of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20? 3. What is the 3d power of each of the above? The 4th power? The 5th power? The 6th power? PRINCIPLES. 1. Every composite number is equal to the product of its factors. 2. A factor of a number is a factor of any number of times that number. 108. To form composite numbers out of any factors. 1. Form a composite number out of 4, 9, and 5. SOLUTION.—A composite number formed out of the factors, 4, 9, and 5, is equal to their product, which is 4x9x5=180 180. OPERATION. WRITTEN EXERCISES, Form composite numbers out 2. Of 5, 6, 7, and 8. Ans. 1680. 3. Of two 2's, and 7. Ans. 84. 4. Of three 3's, four 2's, and two 5's. Ans. 10800. 5. Find a number consisting of four 5's. Ans. 625 6. Find the fifth power of 3, of 4, of 7. Ans. 243; 1024; 16807. 7. Form a composite number out of the first four prime numbers after unity. Ans. 210. 8. Form a composite number out of all the prime numberg between 11 and 29. Ans. 96577. 9. Form all the composite numbers you can out of 2, 3, 5, and 7. Ans. 6; 10; 14; 15; 21; etc. 10. Form all the composite numbers you can out of 2, 3, 5, 7, and 11. Ans. 6; 10; 14; 22; 15; 21; etc. 11. Find a composite number consisting of three factors, the first being 2, the second being twice as great, and the third three times as great. Ans 48. DIVISIBILITY OF COMPOSITE NUMBERS. 109. Composite Numbers can be divided by the factors which produce them. 110. The Factors of many composite numbers may be seen by inspection from the following principles : PRINCIPLES. 1. A number is divisible by 2 when the right hand term is zero or an even digit. For, the number is evidently an even number, and all even numbers are divisible by 2. 2. A number is divisible by 3 when the sum of the digits is divisible by 3. This may be shown by trying several numbers, and, seeing that it is true with these, we infer that it is true with all. A rigid demonstration is too difficult for this place. 3. A number is divisible by 4 when the two right hand terms are ciphers, or when the number they express is divisible by 4. If the two right hand terms are ciphers, the number equals a number of hundreds, and since 100 is divisible by 4, any number of hundreds is divisible by 4. If the number expressed by the two right hand digits is divisible by î, the number will consist of a number of hundreds plus the number expressed by the two right hand digits (thus 1232=1200+32); and since both of these are divisible by 4, their sum, which is the number itself, is divisible by 4. 4. A number is divisible by 5 when its right hand term. is 0 or 5. When the unit figure is 0, the last partial dividend must be 0, 10, 20, 30, or 40, each of which is divisible by 5. When the unit figure is 5, the last partial dividend must be 15, 25, 35, or 45, each of which is divisible by 5. Therefore, etc. 5. A number is divisible by 6 when it is even, and the sum of the digits is divisible by 3. Since the number is even it is divisible by 2, and since the sum of the digits is divisible by 3 the number is divisible by 3, and since it contains both 2 and 3, it will contain their product, 3X2, or 6. 6 A number is divisible by 8 when the three right hand terms are ciphers, or when the number expressed by them is divisible by 8. If the three right hand terms are ciphers, the number equals 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 number expressed by the three right hand digits (thus 17368317000+368) and since both of these parts are divisible by 8, their sum, which is the number itself, is divisible by 8. 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 rigidiy 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 thie difference is 0. 3 These two principles are rather curious than useful. For their de 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 prime 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 ind 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 numbers? Ans. Factoring. 11. How then may we define the subject of Factoring? |