5.1 Number Theory 221 5. A natural number that is divisible by a number other than itself and 1 is known as a(n) __________ number. Composite 6. The smallest natural number that is divisible (without remainder) by each number in a set of numbers is known as the least common __________ of the set of numbers. Multiple 7. The largest natural number that divides (without remainder) each number in a set of numbers is known as the greatest common __________ of the set of numbers. Divisor 8. Prime numbers of the form 2 1, n − where n is a prime number, are known as __________ prime numbers. Mersenne 9. A hypothesis that has not been proved or disproved is known as a(n) __________. Conjecture 10. Goldbach’s conjecture states that every even number greater than or equal to 4 can be written as the sum of two (not necessarily) distinct __________ numbers. Prime Practice the Skills In Exercises 11 and 12, use the sieve of Eratosthenes to determine the prime numbers less than or equal to the given number. 11. 100 * 12. 150 * In Exercises 13–26, determine whether the statement is true or false. Modify each false statement to make it a true statement. 13. 7 is a factor of 42. True 14. 45 is a factor of 5. False. 5 is a factor of 45. 15. 6 is a multiple of 36. False. 36 is a multiple of 6. 16. 72 is a multiple of 9. True 17. 3 is a divisor of 15. True 18. 32 is a divisor of 4. False. 4 is a divisor of 32. 19. 8 64. | True 20. 39 3. | False. 3 39. | 21. 4 is a divisor of 84. True. 22. 54 is a divisor of 9. False. 9 is a divisor of 54 In Exercises 33– 40, determine the prime factorization of the number. 33. 20 2 5 2 ⋅ 34. 48 3 42 ⋅ 35. 120 2 3 5 3 ⋅ ⋅ 36. 315 3 5 7 2 ⋅ ⋅ 37. 1336 2 167 3 ⋅ 38. 3190 2 5 11 29 ⋅ ⋅ ⋅ 39. 21,888 2 3 4 19 3 2 2 ⋅ ⋅ ⋅ 40. 33,480 2 3 5 31 3 3 ⋅ ⋅ ⋅ In Exercises 41– 44, determine the greatest common divisor (GCD). 41. 18 and 24 6 42. 20 and 35 5 43. 32 and 224 32 44. 98 and 224 14 In Exercises 45– 48, determine the least common multiple (LCM). 45. 12 and 30 60 46. 20 and 42 420 47. 96 and 212 5088 48. 48 and 110 2640 In Exercises 49–54, determine (a) the greatest common divisor (GCD) and (b) the least common multiple (LCM). 49. 10 and 25 a) 5 b) 50 50. 12 and 21 a) 3 b) 84 51. 12 and 32 a) 4 b) 96 52. 16 and 28 a) 4 b) 112 53. 75 and 120 a) 15 b) 600 54. 120 and 240 a) 120 b) 240 Problem Solving 55. Landscaping Steve cuts his grass every six days and weeds his flower beds every 10 days. If Steve cuts his grass and weeds his flower beds on August 1, how many days will it be before he cuts his grass and weeds his flower beds on the same day again? 30 days 56. U.S. Senate Committees The U.S. Senate consists of 100 members. Senate committees are to be formed so that each of the committees contains the same number of senators and each senator is a member of exactly one committee. The committees are to have more than 2 members but fewer than 50 members. There are various ways that these committees can be formed. m U.S. Capitol Building *See Instructor Answer Appendix Karl Kost/Alamy Stock Photo 23. If a number is not divisible by 5, then it is not divisible by 10. True 24. If a number is not divisible by 10, then it is not divisible by 5. False; a number may be divisible by 5 if it is not divisible by 10. 25. If a number is divisible by 2 and 3, then the number is divisible by 6. True 26. If a number is divisible by 3 and 5, then the number is divisible by 15. True In Exercises 27–32, determine whether the number is divisible by each of the following numbers: , , , , , , , 2 3 4 5 6 8 9 and . 10 27. 1155 3 and 5 28. 563,240 2, 4, 5, 8, and 10 29. 72,216 2, 3, 4, 6, 8, and 9. 30. 170,820 2, 3, 4, 5, 6, 9, and 10. 31. 1,882,320 2, 3, 4, 5, 6, 8, and 10. 32. 3,941,221 None of the numbers.
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