# Solutions: Number Theory

#### Factors and Greatest Common Factors

 Exercise Problem Solution 1 Find the greatest common factor of 18 and 36. Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 GCF = 18 2 Find the greatest common factor of 30 and 48. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 GCF = 6 3 Find the greatest common factor of 42 and 56. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 GCF = 14 4 Find the greatest common factor of 16, 48 and 72. Factors of 16: 1, 2, 4, 8, 16 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 GCF = 8 5 One rectangular garden has an area of 18 square meters. Another rectangular garden has an area of 81 square meters. What is the largest possible dimension common to both gardens? Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 81: 1, 3, 9, 27, 81 GCF = 9

#### Multiples and Least Common Multiples

 Exercise Problem Solution 1 Find the least common multiple of 9 and 10. Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, ... Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ... LCM = 90 2 Find the least common multiple of 14 and 42. Multiples of 14: 14, 28, 42, 56, 70, 84, ... Multiples of 42: 42, 84, ... LCM = 42 3 Find the least common multiple of 18 and 30. Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ... Multiples of 30: 30, 60, 90, 120, 150, 180, ... LCM = 90 4 Find the least common multiple of 8, 9, and 12. Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ... Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ... Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, ... LCM = 72 5 Mrs. Hernandez waters one of her plants every 10 days and another plant every 14 days. If she waters both plants today, when is the next time both plants will be watered on the same day? Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, ... Multiples of 14: 14, 28, 42, 56, 70, 84, 98, ... LCM = 70 (Answer: in 70 days)

#### Prime and Composite Numbers

 Exercise Problem Solution 1 Each of the following numbers is composite EXCEPT: 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 32 and 39 30 and 34 31 and 37 All composite. 31 and 37 2 The prime numbers between 40 and 49 are: 42, 43 and 47 41, 43 and 47 43, 45 and 47 None of the above. 41, 43 and 47 3 The prime numbers between 50 and 59 are: 53 and 59 51 and 59 53 and 57 None of the above. 53 and 59 4 The prime numbers between 60 and 69 are: 63 and 69 61 and 67  60 and 65 None of the above. 61 and 67 5 The prime numbers between 20 and 69 are: 21, 23, 29, 31, 37, 41, 43, 47, 53, 63 and 69 23, 29, 31, 33, 37, 41, 43, 47, 59, 61 and 67 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 and 67 None of the above. 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 and 67

#### Divisibility Tests

 Exercise Problem Solution 1 The number 477 is divisible by each of the following numbers EXCEPT: 3, 6, 9 3 9 6 None of the above. 6 2 The number 348 is divisible by each of the following numbers EXCEPT: 2, 3, 4, 5, 6 6 5 4 None of the above. 5 3 If a number is divisible by 9, then it is also divisible by which number? 3 6 2 None of the above. 3 4 Which number is divisible by 8? 23,624 39,126 120,458 None of the above. 23,624 5 Which number is divisible by 6? 213 468 621 None of the above. 468

#### Exponents

 Exercise Problem Solution 1 Write 45 in standard form. 45 = 4 x 4 x 4 x 4 x 4 = 1024 2 Write 54 in standard form. 54 = 5 x 5 x 5 x 5 = 625 3 What is 500,000,000 raised to the zero power? 1; Any number (except 0) raised to the zero power is always equal to 1. 4 What is 237 raised to the first power? 237; Any number raised to the first power is always equal to itself. 5 The number 81 is 3 raised to which power? 81 = 3 x 3 x 3 x 3, or 3 raised to the fourth power.

#### Patterns and Exponents

 Exercise Problem Solution 1 The numbers 1, 5, 25, 125, 625, ... are each powers of what number? 5 2 In Exercise 1, what is the next number in the list? (Do not enter commas in your answer.) 3125 3 The numbers 1, 6, 36, 216, 1,296, ... are each powers of what number? 6 4 10,000,000,000,000 is 10 raised to what power? 13 5 If 14 is equal to 1, then what is 1100? 1

#### Practice Exercises

 Exercise Problem Solution 1 Find the greatest common factor of 28 and 72. Factors of 28: 1, 2, 4, 7, 14, 28 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 GCF = 4 2 Find the least common multiple of 15 and 25. Multiples of 15: 15, 30, 45, 60, 75, ... Multiples of 25: 25, 50, 75, ... LCM = 75 3 Name one prime number between 80 and 89. 83 or 89 4 The number 91 is composite. Name a number other than 1 and itself that is a factor of 91. 7 or 13 5 Which number is divisible by 3? 49, 87, 103, 119 87 since the sum of its digits is 15; and 15 is divisible by 3. 6 Which number is divisible by 8? 1821, 1496, 1246, 1586, 1634 1496 since 496 is divisible by 8. 7 Write fourteen squared in standard form. 142 = 14 x 14 = 196 8 Write nine cubed in standard form. 93 = 9 x 9 x 9 = 729 9 The numbers 1, 8, 64, 512, ... are each powers of 8. What is the next number in the list? 512 x 8 = 4096 10 If five to the fourth power is 625, then what is five raised to the fifth power? 55 = 54 x 5 = 625 x 5 = 3125

#### Challenge Exercises

 Exercise Problem Solution 1 One rectangular rug has an area of 24 square yards. Another rectangular rug has an area of 90 square yards. If you examine all whole-number dimensions each rug can have, what is the largest possible dimension common to both rugs? Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 GCF = 6 (Answer: 6 yards) 2 A school hallway has a long row of lockers. Every sixth locker contains a package of chewing gum, every eighth locker contains a mirror, and every ninth locker contains a hockey stick. What is the first locker that contains all three items? Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ... Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, ... Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, ... LCM = 72 (Answer: Locker 72) 3 Name one prime number between 70 and 79. 71 or 79 4 The number 279 is composite. Name a number other than 1 and itself that is a factor of 279. 3, 9, 31 or 93 5 Which number is divisible by 4? 386, 418, 568, 694 568 6 Which number is divisible by 6? 496, 246, 589, 634 246 7 Write twelve squared in standard form. 122 = 12 x 12 = 144 8 Write eight cubed in standard form. 83 = 8 x 8 x 8 = 512 9 The numbers 1, 7, 49, 343, ... are each powers of 7. What is the next number in the list? 343 * 7 = 2401 10 If two raised to the eleventh power is 2,048, then what is two raised to the twelfth power? (Do not enter commas in your answer.) 212 = 211 x 2 = 2048 x 2 = 4096