Properties of Numbers. DEFINITIONS. ART. 81.-An Integer is a whole number; as 1, 5, 6, 11. Integers are either Even or Odd. ART. 82.-An Even Number is a number that is exactly divisible by 2; as 4, 6, 8, 10, 14. ART. 83.–An Odd Number is a number that is not exactly divisible by 2; as 3, 5, 7, 9, 11. Numbers are either Composite or Prime. ART. 84.—A Composite Number is one that may be produced by multiplying together two or more numbers each greater than 1; as 9, 10, 15, 16, 24. ART. 85.—A Prime Number is one that cannot be produced by multiplying together two or more numbers each greater than 1; as 7, 11, 13, 19, 29. Numbers are prime to each other when no number greater than 1 exactly divides them; as 5, 7; 4 and 9. ART. 86.—A Prime Factor is a factor that is a prime number; thus, 2, 3, and are the prime factors of 42. ART. 87.-A Common Divisor of two or more numbers is any number greater than 1 that exactly divides each of them; thus, 4 is a common divisor of 12, 20, 24; and 5 is a common divisor of 15, 25, 30, 35. ART. 88.—The Greatest common Divisor of two or more numbers is the greatest exact divisor of each of them; thus, 9 is the greatest common divisor of 18, 27, and 36; and 15 is the greatest common divisor of 30, 45, 60, 90. ART. 89.-A Multiple of a number is any number of times that number; thus, 6, 9 and 12 are multiples of 3. A Common Multiple of two or more given numbers is any number exactly divisible by each of them; thus, 24 is a common multiple of 4, 6, 8 and 12. ART. 90.—The Least Common Multiple of two or more numbers is the least number that is exactly divisible by each of them; thus, 30 is the least common multiple of 2, 3, 5, 10 and 15. Factors. ORAL EXERCISES. ART. 91.–1. What are the factors of 9; Solution. The factors of 9 are 3 and 3, because 3 times 3 is 9. What are the factors : 2. Of 15 ? 6. Of 34 ? 10. Of 48 ? 3. Of 16 ? 7. Of 38 ? 11. Of 60 ? 12. Of 175 ? 13. Of 100? ART. 92.—Principles. 1. Every prime factor of a number is an exact divisor of that number. 2. Every composite number equals the product of all its prime factors. 14. What are the prime factors of 16 ? Solution. The prime factors of 16 are 2, 2, 2 and 2, because they are the only prime numbers whose product is 16. What are the prime factors : 25. Of 35 ? 26. Of 39 ? 17. Of 15 ? 22. Of 30 ? 27. Of 44 ? 18. Of 20 ? 23. Of 32 ? 28. Of 49 ? 19. Of 21? 24. Of 33 ? 29. Of 57 ? 30. Of what number are 3, 2 and 3 the prime factors ? 2, 3, 2 and 3? 3, 5 and 2? 3, 2 and 7? WRITTEN EXERCISES. 20240 2)120 ART. 93.-1. What are the prime factors of 480 ? Process. Analysis.—Dividing 480 by the prime number 2 gives 240 for a quotient ; dividing again by 2, gives 2)480 120 ; dividing again by 2, gives 60 ; dividing again by 2, gives 30 ; dividing this by the prime 260 number 2, gives 15 ; and dividing this by the 2)30 prime number 3, gives the prime number 5 as the 3)15 quotient. Since we have divided only by prime 5 numbers, and since our last quotient is a prime number, it is evident that the several divisors and the last quotient inust form the prime factors of 480. Therefore, 2, 2, 2, 2, 2, 3 and 5 are the prime factors of 480. ART. 94.–Rule for finding the Prime Factors of a Number. Divide the number by the least prime number that will divide it; divide the quotient thus obtained by the least prime number that will divide it, and continue the process until the quotient is a prime number ; the several divisors and the last quotient are the prime factors required. What are the prime factors : 2. Of 21 ? 9. Of 88 ? 3. Of 24 ? 10. Of 81 ? 4. Of 36 ? 11. Of 96 ? 5. Of 42 ? 12. Of 144 ? 6. Of 48 ? 13. Of 275 ? 7. Of 175 ? 14. Of 348 ? 8. Of 84 ? 15. Of 382 ? 16. Of 395 ? 23. Of 1284 ? 26. Of 31254 ? 29. Of 20482 ? 24. Of 3216 ? 27. Of 48762 ? 30. Of 39516 ? 25. Of 4124 ? 28. Of 51946 ? 31. Of 42856 ? 32. Name three numbers and find their prime factors. Divisors. ORAL EXERCISES. ART. 95.-1. What is the common divisor of 9 and 12 ? Solution.—Three is the common divisor of 9 and 12, because it is exactly contained in each of them. 2. What are the common divisors of 12 and 16 ? 20 and 25 ? 3. What are the common divisors of 18 and 21 ? 24 and 28 ? ART. 96.—Principles. 1. The product of all the prime factors common to two or more numbers is their greatest common divisor. 2. The greatest common divisor of two numbers is a divisor of their sum. 3. The greatest common divisor of two numbers is a divisor of their difference. WRITTEN EXERCISES. ART. 97.—1. Find the greatest common divisor of 84 and 90. FIRST METHOD. Analysis. — By inspecting prime factors of which 90 and 84 90 2 x 3 x 3 x 5 are composed, we observe that 2 and 3 are the only prime factors common to both numbers. Therefore, 2 x 3 or 6 must be the greatest common divisor of 84 and 90. ART 98.—Rule for finding the Greatest Common Divisor, Resolve each number into its prime factors ; the product of the prime factors common to all the numbers is the greatest common divisor. the i SECOND METHOD. 2. Find the greatest common divisor of 72 and 96. Process. Analysis.—The greatest common divisor of two or more numbers cannot be greater 72)96(1 than the least number. 72 is not an exact 172 divisor of 96, since there is a remainder of 24. 24)72(3 72 The difference (Prin. 3) contains the great est common divisor. 24 is an exact divisor of 72, and is therefore the greatest common divisor of 72 and 96. ART. 99.—Rule for finding the Greatest Common Divisor. Divide the greater number by the less, and, if there be a remainder, divide the divisor by it, and so continue to divide the last divisor by the last remainder until nothing remains. The last divisor is the greatest common divisor. If there are more than two numbers, find the greatest common clivisor of two of them ; then of this divisor, and a third number, and so on. The last divisor is the greatest common divisor required. Find by the first method the greatest common divisor of: 3. 20 and 66. 8. 64 and 114. 4. 42 and 82. 9. 70 and 180. 5. 36 and 78. 10. 150 and 275. 6. 120 and 132. 11. 126 and 264. 7. 125 and 175, 12. 640 and 960. 13. Find by the second method the greatest common divisor of 360, 484 and 684. 14. Find by the second method the greatest common divisor of 280, 640 and 1728. 15. Find by the second method the greatest common divisor of 542, 1084 and 1626.. |