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1 Answers Investigation Applications. a. Divide 24 by 2 to see if you get a whole number. Since 2 * 2 = 24 or 24, 2 = 2, 2 is a factor. b. Divide 29 by 7 to see if the answer is a whole number. Since 29, 7 = , 7 is not a factor of a. 4 b. 9 c. 8 d a. 2 has the most factors; and 9 are the factors of 9;, 3, 7, and 2 are the factors of 2; and 23 are the factors of 23;, 5, and 25 are the factors of 25. b. Answers will vary. Possible answer is 36;, 2, 3, 4, 6, 9, 2, 8, and 36 are the factors of 36. c. Answers will vary. Possible answer is 8;, 2, 3, 6, 9, and 8 are the factors of a. 6; Since the result is a whole number, 4 is a factor of 84. b. 5.6; Since the result is not a whole number, 5 is not a factor of a. Yes; 8, 6 = 3 b. No; 6, 8 = c 6. 2, 8, 6 are divisors of a. n = 8 b. n = 2 c. n = 2 d. n = 20 e. n = 3 8. a. (See Figure.) b. Keiko picked 28. Cathy scored 2 points for 7 and 4, the factors of 28 that were not already circled. Here is the game board at this stage in the game: (See Figure 2.) c. Cathy picked 8 or 27. Keiko scored 9 points by circling 9 (the factor of 8 or 27 that is not already circled). d. Cathy scores 30 points for 5, 0, and 5. (the factors of 30 that are not already circled). e. The only numbers remaining with uncircled factors are 22 () and 26 (3). Cathy should choose 26 because she will score 3 more points than Keiko. 9. Check every number beginning with until you begin to get the same factors over again. With each small number factor you find, you will find a second factor when you divide. Factors of 0 are:, 2, 5, 0,, 22, 55, 0. I know I have found all of the factors because I checked all the numbers from to 0, and then started getting repeats. 0. a. 30 is the least possibility. Others are all multiples of 30. Common factors of any two such numbers must include, 2 * 3 = 6, 2 * 5 = 0, 3 * 5 = 5 and 2 * 3 * 5 = 30 in addition to 2, 3, and 5 b. 8 is the least possibility. Others are all multiples of 8. Common factors of any three such numbers must include in addition to 2, 4, and 8. Figure Figure 2 Prime Time Investigation

2 Answers Investigation. a. In the Factor Game, your opponent scores points for proper factors of the number you choose. The only proper factor of prime numbers, such as 2, 3, or 7, is. b. Some numbers, such as 2, 20, and 30, have many proper factors that would give your opponent more points. (Some numbers, such as 9, 5, and 25, have fewer factors and would give you more points.) 2. a. i. 3, 6, 9, 2, 5, 8, 2, 24, 27, 30. ii. Multiples of 3; 3, 6, 9, 2, 5, 8, 2, 24, 27, and 30. iii. 2, 2, 30, and 20 will all appear in the sequence, because they are all multiples of 3. b. i. 7, 4, 2, 28, 35, 42, 49, 56, 63, 70. ii. Multiples of 7 iii. 2 and 20 will appear in the sequence, because they are multiples of 7. c. i. 6 * = 6; 6 * 2 = 2; 6 * 3 = 8; 6 * 4 = 24; 6 * 5 = 30; 6 * 6 = 36; 6 * 7 = 42; 6 * 8 = 48; 6 * 9 = 54; and 6 * 0 = 60. ii. Multiples of 6 iii. 2, 30, and 20 will appear in the sequence, because they are multiples of a. (See Figure 3.) b. 3, 37, 4, 43, and 47 are prime. c. 36 and 49 are square numbers. d. 47; it is the greatest prime number. e. 48; The second player gets 76 points, which is 28 more points than the first player. Figure 3 First Move Proper Factors My Score Opponent's Score , 2, 4, 8, , 3, , 2, , 5, , 2, 3, 4, 6, 9, 2, , 2, , 3, , 2, 4, 5, 8, 0, , 2, 3, 6, 7, 4, , 2, 4,, , 3, 5, 9, , 2, , 2, 3, 4, 6, 8, 2, 6, , Prime Time 2 Investigation

3 Answers Investigation 4. a. Move the paper clip from 6 to make the products 5 *, 5 * 2, 5 * 3, 5 * 4, 5 * 5, 5 * 7, 5 * 8, and 5 * 9; move the paper clip from the 5 to make the products 6 *, 6 * 2, 6 * 3, 6 * 4, 6 * 6, 6 * 7, 6 * 8, and 6 * 9. b. Moving the paper clip from the 6 to the 3, 4, or 9, makes 5, 20, or 45, respectively; moving the paper clip from the 5 to the 7 makes 42. c. Moving the paper clip from the 5 to the 7 makes 6 * 7, which is 42. d. Possible answer: Move the 5 to the 7 to get 42; this blocks the opponent and gets 3 in a row. 5. a. 3 * = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 2; 3 * 5 = 5; 3 * 6 = 8; 3 * 7 = 2; 3 * 8 = 24; 3 * 9 = 27. So you can get a 3, 6, 9, 2, 5, 8, 2, 24 and 27, which are all multiples of 3. b. 3 * = 33; 3 * 3 = 39; 3 * 7 = 5; 3 * 9 = 57; 3 * 20 = 60; and many others. c. There are infinitely many multiples of a. 2 and 9; or 3 and 6 b. and 8 7. a. 36 can be found on the product game by 6 * 6 or 4 * is composite. b. 5 can be found on the product game by only * 5. 5 is prime. c. 7 can be found on the product game by only * 7. 7 is prime. d. 9 can be found on the product game by * 9 or 3 * 3. 9 is composite. 8. Since the numbers on the game board are multiples of the numbers given as possible factors, you could argue in support of Sal s position. The Product Game is more specific, because it implies the result of multiplying the two numbers together, while there are infinite number multiples of two chosen numbers. 9. a. 2 b. No. All other even numbers have 2 as a factor 2, in addition to and themselves. 20. a. 2, 3, and 7 b. 2 is missing. 2. a. 3, 5, 6, and 7 b. 25 is missing. 22. dimensions: * 24, 2 * 2, 3 * 8, 4 * 6, 6 * 4, 8 * 3, 2 * 2, 24 * factor pairs:, 24; 2, 2; 3, 8; 4, dimensions: * 32, 2 * 6, 4 * 8, 8 * 4, 6 * 2, 32 * factor pairs:, 32; 2, 6; 4, dimensions: * 48, 2 * 24, 3 * 6, 4 * 2, 6 * 8, 8 * 6, 2 * 4, 6 * 3, 24 * 2, 48 * factor pairs:, 48; 2, 24; 3, 6; 4, 2; 6, dimensions: * 45, 3 * 5, 5 * 9, 9 * 5, 5 * 3, 45 * factor pairs:, 45; 3, 5; 5, dimensions: * 60, 2 * 30, 3 * 20, 4 * 5, 5 * 2, 6 * 0, 0 * 6, 2 * 5, 5 * 4, 20 * 3, 30 * 2, 60 * factor pairs:, 60; 2, 30; 3, 20; 4, 5; 5, 2; 6, dimensions: * 72, 2 * 36, 3 * 24, 4 * 8, 6 * 2, 8 * 9, 9 * 8, 2 * 6, 8 * 4, 24 * 3, 36 * 2, 72 * factor pairs:, 72; 2, 36; 3, 24; 4, 8; 6, 2; 8, a. Prime numbers have only two factors, and itself. Examples: 2, 3, 5, 7,, two 30. B b. Square numbers have odd numbers of factors. Examples: 4, 9, 6, 25, 36,... c. No. Because prime numbers have two factors (from a) and square numbers have an odd number of factors (from b), you could never have a prime number that is also square. Prime Time 3 Investigation

4 Answers Investigation 3. His number is 25. His number must be a square number because it has an odd number of factors. 6 has five factors and 36 has nine factors fans in row, 50 fans in 2 rows, 25 fans in 4 rows, 20 fans in 5 rows, 0 fans in 0 rows, 5 fans in 20 rows, 4 fans in 25 rows, 2 fans in 50 rows, or fan in 00 rows. Answers about which arrangement to choose will vary. Sample: I would rather have one long banner that wraps around part of the stadium, so I would choose 00 fans in one row. Sample: I would rather have a big square that you could see on TV, so I would choose ten fans in ten rows. Connections hours * 60 minutes = 300 minutes; hour = 330 minutes. 330, 2 = Therefore, they can play 27 games. 35. B; Carlos read = 86 pages the first part of the week. He had = 58 pages left for Thursday and Friday. Since he read the same number of pages each day, 58, 2 = 29. Carlos read 29 pages on Thursday. 36. a. For n , the possible answers of n are 0,,2,3,4,5,...,45,46. b. For 3n 6 50, the possible answers of n are 0,,2,3,4,5,...,5, has many factors, so it can be divided into many equal parts. Since 23 is prime, it cannot be subdivided. The only proper factors of 25 are and 5, so it can only be subdivided into 5 groups of a. * 64, 2 * 32, 4 * 6, 8 * 8, 6 * 4, 32 * 2, and 64 *. b. Answers will vary as they did in Exercise Because 60 has many factors, and 59 and 6 do not. 39. a. Various answers; for example, group sizes, 2, 4, 5, 6, 3, 2, 2, 3, and 2. The goal is to find 0 numbers whose sum is 30. b. Group sizes 3, 3, 3, 3, 3, 3, 3, 3, 3, and 3. If she does not have ten groups, she could have group of 30 students, 2 groups of 5 students each, 5 groups of 6 students each, 6 groups of 5 students each, 0 groups of 3 students each, 5 groups of 2 students each, or 30 groups of student each. c. In part (a), the sum of the numbers in the ten groups must be 30. In part (b), we are considering the factors of days Extensions 4. a = 7 b = 57 c. 97, because it is the largest prime less than The numbers that have two odd digits (Clue 2) and give a remainder of 4 when divided by 5 (Clue ) are 9, 39, 59, 79, and 99. Of these numbers, 9 is the only one with digits that add to 0 (Clue 3). The number is 9. Prime Time 4 Investigation

5 Answers Investigation 43. a. (See Figure 4.) b. Abundant means more than enough, which is appropriate since the sum of an abundant number s factors is more than the number. Deficient means not enough, which is appropriate since the sum of a deficient number s factors is less than the number. Perfect means exactly right, which is appropriate because the sum of a perfect number s factors is equal to the number. c. abundant d. deficient 44. a = 5; your opponent scores one fewer point. b. + 2 = 3; your opponent scores one fewer point. c. 2, 8, and 32. (Any power of 2 that is less than 49.) The name near-perfect fits because the sum of the factors is less than the total needed for the number to be perfect. d. 64 and = 63 and = Some possible answers: 3 * 0 * 0; 2 * 3 * 50; many more. 46. a. 2 * 294 = 588 b., 588; 2, 294; 3, 96; 4, 47; 6, 98; 7, 84; 2, 49; 4, 42; 2, 28 c. There are twice as many rectangles as factor pairs because you can reverse each factor pair to find another rectangle: * 588; 2 * 294; 3 * 96; 4 * 47; 6 * 98; 7 * 84; 2 * 49; 4 * 42; and 2 * 28; then reverse: 28 * 2; 42 * 4; 49 * 2; 84 * 7; 98 * 6; 47 * 4; 96 * 3; 294 * 2; and 588 *. 47. a. 0 * 0 b. 9 * 9 c. If students know of the square root, this is where the factors reverse order. Another way to explain this is to find the factor pair that makes a square. d. The factor pairs reverse order at the square root of the original number. Geometrically, if you use the factor pairs to build rectangles, the rectangle that is closest to a square will be the last factor pair before the pairs reverse order. e. Yes, (similar to d). Because there are only two possible ways to write a prime number as a product of its factors, the turn around occurs after the first pair. Figure , 3, 4, 5, 7, 8, 9, 0,, 3, 4, 6, 7, 9, 2, 22, 23, 25, 26, 27, Abundant Deficient Perfect Prime Time 5 Investigation

6 Answers Investigation 48. Students might notice that for square numbers the last factor pair is the one that uses the square root, though they may not use that vocabulary word. Extending this idea to the non-square numbers, they may say that looking for a square near the number being investigated helps; thus, 66 is close to 8 8, and 7 and 8 are not factors of 66, so there is no point in looking further than 6. (See Figure 5.) 49. a. Samples: 4 = 2 * 2, 9 = 3 * 3,... b. Samples: 8 = 2 * 4, 2 = 2 * 6,... c. Samples: 6 = 4 * 4, 24 = 4 * 6,... d. No; prime numbers have only two factors, and itself. They couldn t be the product of two prime numbers, a prime number and another composite number, or two composite numbers. Figure 5 Number Last Factor Pair Prime Time 6 Investigation

7 Answers Investigation 2 Applications. 24, 48, 72, and 96; the LCM is , 30, 45, 60, 75, and 90; the LCM is ; the LCM is ; the LCM is ; the LCM is ; the LCM is , 84; the LCM is ; the LCM is a. Possible answers: 3, 5; 8, 9; 7, b. They have no common factors except. 0. Possible answers: 2, 5;, 0. Possible answers: 4, 9; 8, Possible answers: 4, 5; 2, 5 3. Possible answers: 3, 35; 7, 5 4. a. Twenty-four -hour shifts; twelve 2-hour shifts; eight 3-hour shifts; six 4-hour shifts; four 6-hour shifts; three 8-hour shifts; two 2-hour shifts, and one 24-hour shift. These are all factors of 24. b. 45 seconds, which is the LCM of 9 and days 6., 2, 3, and 6; the GCF is ; the GCF is. 8., 3, 5, and 5; the GCF is ; the GCF is. 20., 7; the GCF is 7. 2., 5; the GCF is , 2; the GCF is , 3, 7, 2; the GCF is D 25. F Prime Time 26. D 27. a. 2 packages of hot dogs and 3 packages of buns; hot dog and bun b. 0 packages of hot dogs and 5 packages of buns; 4 hot dogs and 4 buns members; each member gets cookie and 2 carrot sticks. 0 members; each member gets 2 cookies and 4 carrot sticks. 5 members; each member gets 4 cookies and 8 carrot sticks. 4 members; each member gets 5 cookies and 0 carrot sticks. 2 members; each member gets 0 cookies and 20 carrot sticks. member; the member gets all 20 cookies and 40 carrot sticks. 29. a. Answers will vary. Sample: The Morgan family buys a 2-pack of bottled water and a 24-pack of boxes of raisins. Each person in the family gets the same number of bottles of water and the same number of boxes of raisins. How many people could the Morgan family have? b. Answers will vary. Sample: John eats an apple once a week. Ruth eats an apple every third day. If they both eat an apple today, when will John and Ruth next eat an apple on the same day? c. The Morgan family could have, 2, 3, 4, 6, or 2 people; these numbers are common factors of 2 and 24. John and Ruth will next eat an apple on the same day in 2 days; this problem involves overlapping cycles, so it can be solved with common multiples. 30. Students need to be able to reason proportionally (without knowing that vocabulary) to move from 20 minutes in day to hour in 3 days to 2 hours in 36 days. Julio s watch gains 2 hours in 36 days. Mario s watch gains 2 hours in 2 days. Since 2 is a factor of 36, the watches will next show the correct time together 36 days after Julio and Mario set their watches, or at 9:00 a.m. on the 6th Tuesday. Investigation 2

8 Answers Investigation 2 3. a. 30 students; each student receives 4 cans of juice and 3 packs of crackers because 20 = 30 * 4 and 90 = 30 * 3. b. 8 students; each student receives 5 cans of juice and packs of crackers because 20 = 8 * 5 and 88 = 8 * , 4, 2, and 42 (42 = 2 * 3 * 7 and 6 = 2 * 3.) 33. any odd multiple of 3 b. Aaron s method works when the two numbers in pair don t have any common factors except ; Ruth s method works when the greater of the two numbers in pair is a multiple of the lesser number; Walter s method works when 2 is the greatest common factor of the two numbers in pair. 34. a. Aaron s method does not work for any pair of numbers. For example, the LCM of 4 and 8 is 8, which does not equal 4 * 8, or 32. Ruth s method does not work for any pair of numbers. For example, the LCM of 2 and 7 is 4, which does not equal 7. Walter s method does not work for any pair of numbers. For example, the LCM of 3 and 5 is 5, which does not equal 3 * 5 2, or 5 2. Connections is a factor of is a factor of is a divisor of is a divisor of is a multiple of is a multiple of 9. The product of 7 and 9 is is divisible by is divisible by ; 2 * 4 = ; * 0 = ; 6 * 8 = ; * = ,995,000; multiply 4,995 by,000 for the three factors of a. In 2 hours, the jet will travel 2 * 2 * 60 =,440 kilometers. In 6 hours, the jet will travel,440 * 3 = 4,320 kilometers. b. In 6 hours, the jet will travel 4,320 -,440 = 2,880 kilometers more than in 2 hours. 42. This question also asks students to reason proportionally. (Note: This provides preparation for the next Unit, Comparing Bits and Pieces.) a. 9 * 5 * 7 will be 3 times as great as 3 * 5 * 7, or 35. b. 3 * 5 * 4 will be twice as great as 3 * 5 * 7, or 20. c. 3 * 50 * 7 will be 0 times as great as 3 * 5 * 7, or,050. d. 3 * 25 * 7 will be 5 times as great as 3 * 5 * 7, or a. composite and square (5 * 5 = 25) b. prime c. composite (3 * 7 = 5) d. square ( * = ) c. In 4 hours, the jet would travel twice as many miles as in 2 hours, or 2,880 kilometers. Prime Time 2 Investigation 2

9 Answers Investigation 2 Extensions * 4 * 5 + = 6 (Find the LCM of 2, 3, 4, 5, and 6, and add.) and sometimes; GCF(2, 2) = 2 and GCF(4, 6) = 2, but GCF(6, 2) = 6, GCF(8, 20) = always; every prime number has only two factors, and itself. Therefore, any two different prime numbers have no common factors other than. 48. sometimes; it is only true when the two numbers in the pair are both. 49. always; the greatest factor of any number n is itself. 50. always; the multiples of n are n *, n * 2, n * 3,... The multiples of are *, * 2, * 3,..., * n,... Therefore, LCM(n, ) = n. 5. never; the LCM of any number n greater than and itself is n. Note that the statement is true for n =. 52. sometimes; it is only true when the number doesn t have 3 as a factor. 53. always; the multiples of p are p *, p * 2, p * 3,... The multiples of are *, * 2, * 3,..., * p,... Therefore, LCM(p, ) = p. 54. Answers will vary. Example: The GCF of any composite number c and any prime number p is p; sometimes; it is only true when the composite number in the pair has the factor p. 55. Factors of 30:, 2, 3, 5, 6, 0, 5, and 30. Factors of 36:, 2, 3, 4, 6, 9, 2, 8, and 36. (See Figure.) a. They are common factors of both 30 and 36 (also factors of 6). b. The numbers in the intersection are, 2, 3, and 6. The greatest of these is 6, so 6 is the GCF of 30 and 36. c. The lowest common factor is. is a factor of all whole numbers, so it is in the intersection of Venn diagrams for factors. Figure Factors of Factors of Prime Time 3 Investigation 2

10 Answers Investigation Factors of 20:, 2, 4, 5, 0, and 20. Factors of 27:, 3, 9, and 27. (See Figure 2.) a. There is only one number in the intersection:. It is a factor of both 20 and 27. b. The only number in the intersection is. It is the only common factor of 20 and 27. Since the only factor in common is, it is the GCF. c. In the Venn diagram for 30 and 36, there are several common factors. In the Venn diagram for 20 and 27, only is in the intersection since 20 and 27 have no common factors other than. 57. The multiples of 5 up to 40 are 5, 0, 5, 20, 25, 30, 35, and 40. The multiples of 4 up to 40 are 4, 8, 2, 6, 20, 24, 28, 32, 36, and 40. (See Figure 3.) a. They are multiples of both 5 and 4. b. The least number in the intersection is 20. So, 20 is the LCM of 5 and 4. c. 60, 80, 00, 20, 40, 60, etc. These numbers are all multiples of the LCM, which is 20. There is no greatest common multiple. Figure Factors of 20 Factors of Figure Multiples of Multiples of Prime Time 4 Investigation 2

11 Answers Investigation The multiples of 6 up to 48 are 6, 2, 8, 24, 30, 36, 42, and 48. The multiples of 8 up to 48 are 8, 6, 24, 32, 40, and 48. (See Figure 4.) a. They are multiples of both 6 and of 8. b. The least number in the intersection is 24. So, 24 is the LCM of 6 and 8. c. In the Venn diagram for 5 and 4, only multiples of 5 * 4 are in the intersection. In the Venn diagram for 6 and 8, 24 is in the intersection, but it is not a multiple of 6 * years 60. a. 36 b. 0 c. Eric forgot that multiplication is commutative, e.g., 3 * 4 = 4 * 3. He only needs to know 28 different products. Also, zero times anything is zero, and 3 of the 49 computations involve zero as a factor. (Note: There are only 9 different answers possible using two factors from 0 through 6.) 6. a. 2-year cicadas would meet 2-year predators either every time they emerge or never. The 3-year cicadas would encounter predators every other time they emerge, so they could be better or worse off depending on whether the predator came out on odd or even years. b. The 2-year cicadas would meet one or both types of predators every time they emerge. The 3-year cicadas would meet the 2-year predators every other time they emerge, and the 3-year predators every third time they emerge. This means that it would be 6 cycles, or 78 years, before the 3-year cicadas would have to face both predators again. They are better off than the 2-year cicadas and 6; GCF(4, 6) = 2 and LCM(4, 6) = and 4; LCM(2, 4) = and 8; GCF(4, 8) = and 2; GCF(2, 2) = 2 and LCM(2, 2) = 2. In general, the GCF and LCM are equal if and only if the two numbers in pair are the same and 36; GCF(24, 36) = 2 and LCM (24, 36) = , 0 and 5; GCF(5, 0, 5) = , 6 and 8; LCM(3, 6, 8) = , 3, 5 and 7; GCF(2, 3, 5, 7) =. Figure Multiples of Multiples of Prime Time 5 Investigation 2

12 Answers Investigation 3 Applications. The path is 7 * 5 * 2 * 3 * The path is 3 * 4 * 5 * Mazes will vary. 4. He can give each child 3 cookies, and he will have 2 left for himself * 2 * 3 * * 2 * 3 * 3 * * 5 * 5 * * 5 * * 2 * 2 * 5 * 9. 2 * 2 * 2 * 3 * 3 * * 7 * 3. 2 * 2 * 2 * 3 * = 2 2 * 3 2, 80 = 2 2 * 3 2 * 5, 525 = 3 * 5 2 * 7, 65 = 3 * 5 *, 293 = 293, 760 = 2 3 * 5 * 9, 26 = 2 3 * 3 3, 23 = 3 * 7 *, and 32 = 2 3 * 3 * 3 5. D 6. Jamahl is correct. Possible answer: Consider 26. It has six prime factors: 2 * 2 * 2 * 3 * 3 * 3. Then consider 23. It has three prime factors: 3 * 7 *. 23 is greater than 26, even though it has fewer prime factors. 7. 0, 20, 30, 40, 50, 60, 70, 80, and 90. They are all multiples of H; 2 * 3 * * 3 * 5 = 30, 2 * 3 * 7 = 42, 2 * 3 * = 66, 2 * 3 * 3 = 78, 2 * 5 * 7 = a. 8 * 8 = 64, so Mr. and Mrs. Fisk have 64 grandchildren. b. 64 * 8 = 52, so they have 52 great-grandchildren. c. Expressions may vary. Sample: 8 3 = GCF = 9, LCM = GCF = 5, LCM = GCF = 26, LCM = GCF = 5, LCM = GCF =, LCM =, GCF =, LCM = a. N = 45, 90, or 80. Any other factor of 80 has a common multiple with 2 that is less than 80. b. M = 4 or any multiple of 4 that doesn t have a factor of 3 or , 8, and 625 (the numbers that are a prime number to the 4th power) 29. a. Locker 5 b. Locker 60 c. Locker 504 d. Locker a. Only Student touched both lockers because the locker numbers are relatively prime. b. Students, 2, 5, 7, 0, 4, 35, and 70 touched both lockers 40 and 20. These are the common factors. c. Students, 3, 5,, 5, 33, 55, and 65 touched both lockers 65 and 330. These are the common factors. d. Students, 2, 7, 4, 49, and 98 touched both lockers 96 and 294. These are the common factors. Prime Time Investigation 3

13 Answers Investigation 3 Connections 3. Rosa is correct because the number is not a prime number. Tyee is correct that his string is a longer string of factors, but it is not a string of prime factors for Mathematicians have determined that it is important for a number to be able to be identified by its longest string of factors. If the number were prime, the prime factorization for a number would have to include and could include as a factor any number of times. So a prime factorization would not be the same as the longest string possible without using. 33. a. 00 = 2 2 * 5 2, which has nine factors (, 2, 4, 5, 0, 20, 25, 50, 00). b. 0 is prime and has only two factors (, 0). c. 02 = 2 * 3 * 7, which has eight factors (, 2, 3, 6, 7, 24, 5, 02). d. 03 is prime and has only two factors (, 03). e. Answers may vary. Students may notice that for four consecutive numbers, 2 is a factor of two of the numbers; 3 is a factor of at least one of the numbers; 4 must be a factor of one of the numbers; larger numbers do not necessarily have more factors. 34. These numbers are all multiples of These numbers are all multiples of , 2, 3, 5, 5, and 30 are also common factors. 37. a. 9, 8, 27, 36, 45, 54, 63, 72, 8, 90, and 99 b. 2, 42, 63, and 84 c. 63 d Factor to find that 84 = 8 * 23 and 207 = 9 * is the only common factor other than. If they only earned + a day, they would have to work longer than one month to earn +84 or more. Therefore, Tomas worked 8 days at +23 per day and Sharina worked 9 days at +23 per day. 39. Answers will vary. You might want to have students post some of their stories around the room. Note: The prime factorization of 648 is 648 = 2 3 * Yes. Numbers that end in 0 are multiples of 0. Numbers that are multiples of 0 have both 2 and 5 as factors Since the number is a multiple of 2 and 7 (Clue ), which are relatively prime, the number must be a multiple of 4. The multiples of 4 between 50 and 00 (Clue 2) are 56 = 2 * 2 * 2 * 7, 70 = 2 * 5 * 7, 84 = 2 * 2 * 3 * 7, and 98 = 2 * 7 * 7. Of these numbers, only 70 is the product of three different prime numbers (Clue 3). 43. The factors of 32 are, 2, 4, 8, 6, and 32 (Clue 3). Of these numbers, only, 6 and 32 have digits that sum to odd numbers (Clue 4). and 6 are square numbers (Clue ). Of these two numbers, only 6 has 2 in its prime factorization (Clue 2). The number is 6. Prime Time 2 Investigation 3

14 Answers Investigation A number that is a multiple of 3 (Clue 2) and of 5 (Clue ) must be a multiple of 5. The multiples of 5 that are less than 50 are 5, 30, and 45. Only 30 has 8 factors (Clue 3). 45. Multiples of 5 that don t end in 5 are multiples of 0 (Clue ). The factor string is three numbers long (Clue 2), and two of these are 2 and 5. Since two of the numbers in the factor string are the same (Clue 3), the number is 2 * 2 * 5 = 20 or 2 * 5 * 5 = 50. The number is greater than the seventh prime, 49, so 50 is the number. 46. Answers will vary. You may want to post some of the best student responses for a Problem of the Week. 47. Only Locker 42 meets all three criteria. A good way to approach the problem is to list all the even numbers 50 or less and cross out those that fail to meet each of the other criteria. A student might also start by listing only the numbers divisible by 7 and then apply the other two criteria to those numbers. 48. a. 2, 4, 8, 6, 32, 64, 28, 256, 52 b., This is the only pair of primes that are consecutive numbers. We know that this is the only such pair because 2 is the only even prime number. 50. There are (infinitely) more odd prime numbers. The only even prime is 2. Extensions 5. a. Possible answers: 994 = 2 * 997, 995 = 3 * 5 * 7 * 9, 996 = 2 * 2 * 499, 997 = 997, 998 = 2 * 3 * 3 * 3 * 37, 999 = 999, 2000 = 2 * 2 * 2 * 2 * 5 * 5 * 5, 200 = 3 * 23 * 29, 2002 = 2 * 7 * * 3, 2003 = 2003, 2004 = 2 * 2 * 3 * 67, 2005 = 5 * 40, 2006 = 2 * 7 * 59, 2007 = 3 * 3 * 223, 2008 = 2 * 2 * 2 * 25, 2009 = 7 * 7 * 4, 200 = 2 * 3 * 5 * 67 b. Answers will vary. For example, 996 is not square; it is a multiple of 4 and of 998. It is not prime, and it is even. 52. a. 52 weeks, with extra day if it is not a leap year and 2 extra days if it is a leap year. b. January 8, 5, and 22 c. Monday d. Tuesday e. Your birthday will fall one day later in the week each year, except when leap day (February 29) falls between your birthdays. In that case, your birthday will be two days later in the week. If your birthday is February 29, your birthday will be five days later in the week each time it occurs. Prime Time Answers will vary. Sample: If a number is the least common multiple of several prime numbers, its prime factorization will include only those primes, and no others. 54. The common multiples of 2, 3, 4, 5, and 6 are 60, 20, 80,.... If we add the clue that the box contains fewer than 00 books, the only answer would be a. Barry is correct. The LCM(a,b) = a * b, GCF(a,b). When you multiply a and b together, you will get each of the overlapping prime factors twice. Therefore, if you divide by the GCF, you will get only one set of overlapping factors, along with the factors that do not overlap. b. The product of the LCM and the GCF is equal to the product of the numbers. Investigation 3

15 Answers Investigation Yes. The concept behind David s prediction is that the prime factorization of a number determines all of its factors. If a number is divisible by 2, for instance, then 2 will be a factor of many of the number s factors. In fact, each factor of a number is built up of one or more of the number s other factors. Consider 40. The prime factors of 40 are 2 and 5; the prime factorization is 2 * 2 * 2 * 5, or 2 3 * 5. Consider each of 40 s factors in terms of its own prime factorization: = 2 0 * = 2 * = 2 2 * = 2 3 * = 2 0 * 5 0 = 2 * 5 20 = 2 2 * 5 40 = 2 3 * 5 If we look at all of those numbers in terms of their prime factorizations, we can see every possible arrangement of 2 to the 0,, 2, or 3 power with 5 to the 0 or power (i.e., it s possible to raise the prime factor to the zero power). Prime Time 4 Investigation 3

16 Answers Investigation 4 Applications. An even number minus an even number will be even. Students may use examples, tiles, the idea of groups of two, or the inverse relationship between addition and subtraction. Using an example: 6-4 is 2. Using tiles: For example, if you take away one rectangle with a height of 2 from another rectangle with a height of 2, you will still have a rectangle with a height of 2. Using groups of 2: If you have an even number of objects, you can bundle the number of objects into groups of 2. If you take away some bundles of 2 from a group of bundles of 2, you are still left with bundles of 2. Using the inverse relationship between addition and subtraction: Students may know that if a + b = c, then a = c - b. In this way, the question is asking If c and b are even, is a even or odd? In the equation a + b = c, if the values of b and c are even, then the value of a must also be even, because even + even = even. 2. An odd number minus an odd number is even. If you have a rectangle with one extra square and you take away a rectangle with one extra square, you have taken away the extra square, and you are left with a rectangle with a height of An even number minus an odd number is odd. If you have a rectangle with a height of 2 and you subtract a rectangle with one extra square, you have broken up a pair of squares on the original rectangle and are left with another rectangle with an extra square. 4. An odd number minus an even number is odd. If you have a rectangle with one extra square and you subtract a rectangle with a height of 2, you are left with a rectangle with an extra square. 5. Evens have ones digits of 0, 2, 4, 6, or 8, and they are divisible by 2. Odds have ones digits of, 3, 5, 7, or 9, and they are not divisible by A sum is even if all of the addends are even, or if there is an even number of odd addends. Otherwise, the number is odd * (3 + 6) and (4 * 3) + (4 * 6), total area (4 + 2) * 7 and (4 * 7) + (2 * 7), total area * ( ) and (5 * 3) + (5 * 6) + (5 * 2), total area 55 For Exercises 0 2, the area of the largest rectangle is the sum of the areas of the two smaller rectangles. To find the dimensions of each rectangle, first find a common factor of each pair of numbers. Each Exercise has multiple possible dimensions. 0. (See Figure.) Figure Dimensions of 39 Square Unit Rectangles and Partitions Possible Rectangle Possible Rectangle 2 Small Medium Large Prime Time Investigation 4

17 Answers Investigation 4. (See Figure 2.) 2. (See Figure 3.) 3. 3 * (4 + 6) = 3 * 0 or (3 * 4) + (3 * 6) = * ( ) = 3 * 9 or (3 * 5) + (3 * ) + (3 * 3) = (See Figure 4.) Figure 2 Dimensions of 49 Square Unit Rectangles and Partitions Possible Rectangle Possible Rectangle 2 Small Medium Large Figure 3 Dimensions of 48 Square Unit Rectangles and Partitions Possible Rectangle Possible Rectangle 2 Possible Rectangle 3 Possible Rectangle 4 Small Medium Large Figure Prime Time 2 Investigation 4

18 Answers Investigation 4 5. N * (2 + 6) = 8N or (N * 2) + (N * 6) = 2N + 6N (See Figure 5.) 6. 5 * (N + 2) or (5 * N) + (5 * 2), 5N + 0 N * (30 + 4) = (9 * 30) + (9 * 4) = = 306 (See Figure 6.) 5 5 N 5 2 Figure N N 2 N 6 Figure Prime Time 3 Investigation 4

19 Answers Investigation * 8 = (30 + 5) * (0 + 8) = (30 * 0) + (5 * 0) + (30 * 8) + (5 * 8) = = a. Answers will vary. Possible answer: b. Answers will vary. Possible answer: 6 * 0 c. Answers will vary. Possible answer: 6 * 0 = 6 * (5 + 5) = (5 * 6) + (5 * 6) = a. 90 = = 0(2 + 7) b. 90 = = 9(4 + 6) a. The black number in the lower-righthand square is the sum of the red numbers in the right-most column and also the sum of the red numbers in the bottom row b. The same relationship will hold for any four numbers in the border squares. The black number is always the sum of the red numbers in the right-hand column and the red numbers in the bottom row. (30 + 5) * 8 = (30 * 8) + (5 * 8) = = c. Shalala is correct. The sum of the bottom row is 6(2 + 8) + 3(2 + 8). The sum of the last column is 2(6 + 3) + 8(6 + 3). From the first expression, you can factor (2 + 8) to get (2 + 8)(6 + 3). If you factor (6 + 3) from the second expression, you get (6 + 3)(2 + 8). By the Commutative Property of Multiplication, these two products are equal Note: Some students may write 35 * (20-2) = = 630, although this is not connected to the typical multiplication algorithm. The arithmetic may be easier with these numbers. 24. m = m = 0 2 Prime Time 4 Investigation 4

20 Answers Investigation m = 27. m = (3 + 4) * , 6 * * * 3 = * = * 5 * 3 = * 5-3 = Answers will vary () = 9 or 3 + (2 + 4)() = 9 or ( )() = (2)(4 + ) = (3)( ) = (2)(4) + = = 0 4. a (6 + ) = 25, which is a multiple of 5. b. 4(3) + 6() = 8, which is a factor of a. Answers will vary. Possible answer: 2 = 3 * 7 = 3 * (5 + 2) = 3 * * 2 = b. Answers will vary. Possible answer: 24 = 2 * 2 = 2 * (0 + 2) = 2 * * 2 = c. Answers will vary. Possible answer: 55 = 5 * = 5 * (0 + ) = 5 * * = d. Answers will vary. Possible answer: 48 = 2 * 24 = 2 * (20 + 4) = 2 * * 4 = The student interpreted exponents as multiplying the two numbers. 3 2 is not 6, and 3 3 is not 9. The correct answer is The student performed multiplication before exponentiation; 2 * 3 2 is not 6 2, but 8. The correct answer is The student added before he subtracted; is not 8-2, but The correct answer is The student multiplied before he divided; 24, 6 * 4 is not 24, 24, but 4 * 4. The correct answer is Any number will work. Explanations will vary. Sample: Step : Choose 7. Step 2: = 22 Step 3: (7 + 5) * 2 = 44 Step 4: (7 + 5) * 2-30 = 4 4 = 2 * 7, which is double the original number. Alternatively, (n + 5) * 2-30 = n * = n * 2 which is double the original number. 48. Choose N. Then, ((N * 2 + 6) - 3) = (N * 2, 2) + (6, 2) - 3 = N = N Prime Time 5 Investigation 4

21 Answers Investigation This can be solved algebraically. An area model works as well Let N = the area of a rectangle. N Double it. N Add 6. (See Figure 7.) Divide by 2. (See Figure 8.) Subtract 3. 2 N = N N 2 N = = 2N In Exercises 52 57, each case could be explained by the Distributive Property and knowledge of place value. 52. True. 432 = So 50 * 432 = 50( ) and 50( ) = 50 * * True. 50 * 368 = 50(400-32) = 50 * * 32 N 54. False. If the equation involved subtraction instead of addition, then it would be true. 50 * 800 = (50 *,000) - (50 * 200), since 800 =, False. 90 * 70 = (90 * 30) + (90 * 40); 90 * * 20 and 90 * * 20, so (90 * 30) + (90 * 40) 7 (70 * 20) + (50 * 20). Alternatively, 90 * 70 = 9 * 7 * 00 = 9 * 7 * 5 * * 7 * , though, because is even and 9 * 7 * 5 is odd. 56. False. 50 is not multiplied by the sum ( ). It is added to the product of 400 and True. 6 * 7 = 6(20-3) = 6(20) - 6(3). Each expression is Yes; each expression has a value of a. Mrs. Lee is correct. Because you do multiplication before subtraction, Mrs. Lee s expression will calculate the area of the yard and swing set first, then take the difference of those areas to find the remaining area. b. Mr. Lee is correct. Because you operate in parenthesis first, Mr. Lee s expression will calculate the difference in length first to find the length of the lawn, then multiply by the width to find the area. Figure 7 N N 6 = 2N + 6 Figure 8 2 N 2 N 3 = N + 3 Prime Time 6 Investigation 4

22 Answers Investigation (0 ) (0 ) The expression in expanded form is 7 * 0-7 *. If we simplify within parentheses first, we find the expression is equal to 63: 7(0 - ) = 7 * 9 = 63. If we distribute the 7, we find that the expression is still equal to 63: 7(0 - ) = 7 * 0-7 * = 70-7 = There are 36 # 2 = 432 trading cards and 36 # 2 = 72 stickers (2-3) = 30 # 9 = 270, or Alternatively, students might find the cost for all students 30 # 2 = 360 and then subtract the total discount 30 # 3 = 90 for a total of 270, or Connections 66. A 67. 3, , , , * 5 = 4 * (3 + 2) = (4 * 3) + (4 * 2) 63. Tuesday s high temperature is 3 degrees colder than Sunday s high temperature. Students could use a variable, n, to represent Sunday s temperature. Then Tuesday s temperature can be represented by n + 5-8, which simplifies to n - 3, so Tuesday s high temperature is 3 degrees colder than n, Sunday s temperature. Another method is to choose a few examples to see the relationship. Suppose Sunday s high temperature is 60 degrees. Then Monday s high temperature is 65 degrees, and Tuesday s high temperature is 57 degrees. For any starting amount (Sunday s high temperature), Tuesday s high temperature will be 3 degrees colder. 64. Elijah collected +264, +92 for the school and +72 for his homeroom. Students might calculate the two parts first: 24 # 8 = 92 (school) and 24 # 3 = 72 (homeroom). Solving it this way uses the Distributive Property, because 24() = 24(8 + 3) = 24(8) + 24(3) One way to solve this is to multiply 5(6)(4) = 360. Another number sentence is 5( )4 = 360. In the second equation, students could distribute either the 5 or the 4 to each addend inside the parentheses. 76. a. 32 * 2 is the area of a rectangle with dimensions 32 and 2. The sum of the areas of the bottom two rectangles in Jim s figure is 32 * 2 = (30 + 2) * 2 = 30 * * 2 = = 64, the first partial product in the example. The sum of the areas of the upper two rectangles is 32 * 0 = (30 + 2) * 0 = 30 * * 0 = = 320, the second partial product. By adding these areas together, we get the final product, 384. Prime Time 7 Investigation 4

23 Answers Investigation 4 b. 32 * 8 = (2 + 30) * 8 = (2 * 8) + (30 * 8) = = n must be 2 less than a multiple of 5. 3(n + 2) is a multiple of 5, and 3 is not a multiple of 5, so (n + 2) must be a multiple of 5. Therefore, n must be 2 less than a multiple of 5. (i.e., n must be 3, 8, 3, etc.) n = 0, 2, or 6. The factors of 24 are, 2, 3, 4, 6, 8, 2, and 24. If a number r is a factor of another number q, then the prime factorization of r is a subset of the prime factorization of q. 24 = 2 * 2 * 2 * 3 3(n + 2) = 3 * or 3 * 2 or 3 * 2 * 2 or 3 * 2 * 2 * 2. So n + 2 = or 2 or 2 * 2 or 2 * 2 * 2. (n + 2) cannot equal, because then n would be negative, not a whole number. If n + 2 = 2, then n = 0. If n + 2 = 2 * 2, then n = 2. If n + 2 = 2 * 2 * 2, then n = 6. c. Jim s rectangle: (See Figure 9.), =,620 Basilio s method: 45 * 36 = 45(30 + 6) =, =,620 Figure = = = = 30 Prime Time 8 Investigation 4

24 Answers Investigation n =. The factors of 20 are, 2, 4, 5, 0, and 20. n cannot be 0 because 4(0) + 6 = 6, which is not a factor of 20. n could be, because 4() + 6 = 0, which is a factor of 20. n cannot be 2 because 4(2) + 6 = 4, which is not a factor of 20. n cannot be 3 because 4(3) + 6 = 8, which is not a factor of 20. n cannot be 4 because 4(4) + 6 = 22, which is greater than 20. n cannot be any whole number greater than 4 because any whole number greater than 4 will also result in a number greater than 20. Therefore, n =. The reasoning for Exercise 78 may also be used for Exercise 79, but it results in more instances where N would not be a whole number, and therefore such a method may feel less efficient. Extensions 80. a. Possible answer: Multiples of 2 that are not multiples of 20 include 2, 24, 36, 48, and 72. Multiples of 2 AND 20 include 60, 20, 80, 240, and 300. Multiples of 20 that are not multiples of 2 include 20, 40, 80, 00, and 40. Multiples of neither include, 2, 3, 4, and 5. (See Figure 0.) 8. Either two of the numbers are odd or two are even. If the first number is odd, you will have two odd numbers in the group. If the first number is even, you will have two even numbers. b. The multiples in the intersection are all divisible by 60. Figure 0 Multiples of 2 Multiples of Prime Time 9 Investigation 4

25 Answers Investigation a. Yes, Mirari is correct. Since every third number on the number line is divisible by 3, any three consecutive whole numbers must include one that is divisible by b. No, Gia s statement is not true. For example, = 39, which is not divisible by 6 because it is not even. The sum of three consecutive numbers is divisible by 6 only if the first number is odd. Let n be any whole number. Then 2n + is odd. The sum of the three consecutive numbers beginning with an odd number is (2n + ) + (2n + 2) + (2n + 3) = 6n + 6, which is divisible by 6. If the consecutive numbers began with an even number, then the sum could be represented by 2n + (2n + ) + (2n + 2) = 6n + 3, which has a remainder of 3 when divided by 6. c. Yes, Kim is correct. Any 3 consecutive whole numbers will include a multiple of 3 and a multiple of 2. The product of any group of numbers in which two of the numbers have 2 and 3 as factors must be a multiple of 6 (i.e., the product must be divisible by 6). d. The product of any four consecutive numbers is divisible by 2, 3, 4, 6, 8, 2, and 24. Of the four consecutive numbers, one must be a multiple of 3 and one must be a multiple of 4. In addition to the multiple of 4, there will be one other even number. Since the product has 4 and another even number as its factors, the product will have at least three 2 s in its prime factorization. Therefore, the product is divisible by 2 * 2 * 2 = 8. Since 8 and 3 are relatively prime, 3 * 8, or 24, is also a factor of the product. Therefore, all factors of 24 are also factors of the product. 85. a. Stage 5: = 25 Stage 6: = 36 Stage 7: = 49 Stage 8: = 64 b c + 39 = 20 2 = 400 c. Stage 24; 47. The sum will be 576 in row 24, because 576 = The last number in this row is 47 because 47 is the twenty-fourth odd number. Notice that to find the nth odd number, you can multiply n by 2 and subtract. (This famous pattern is the sum of the consecutive odd numbers: The sum in each row is the square of the number of numbers in the row.) 86. a. 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 0 = or 5 + 5, 2 = 5 + 7, 4 = or 3 +. b. Possible answers: 00 = , 00 = + 89, 00 = , 00 = , 00 = , 00 = c. No, because there are no even prime numbers other than 2. If you added the number 2 to an odd prime, the sum would be odd. 87. a. In general, Boris is not correct, but he is correct for 996. The proper factors of 996 are, 2, 3, 4, 6, 2, 83, 66, 249, 332, and 498. The sum of the factors is,356, which is greater than 996. The proper factors of 975 are, 3, 5, 3, 5, 25, 39, 65, 75, 95, and 325. The sum of these is 76, which is less than 975, so 975 is not an abundant number, even though it has the same number of factors as 996. Prime Time 0 Investigation 4

26 Answers Investigation 4 b. No, there are no square numbers in the list. Square numbers have an odd number of factors, but the numbers in the list all have an even number of factors. Also, students may use their calculators to find that the square root of the least number, 975, is about * 3 = 96, which is less than the least number on the list, and 32 * 32 =,024, which is greater than the greatest number on the list. 88. a. Yes; b. 36 * 5 = (30 + 6) * 5 = (30 * 5) + (6 * 5) = 30 * (0 + 5) + 6 * (0 + 5) = 30 * * * * 5 = = Let N and R be any whole numbers. Then 2N and 2R are even numbers. 2N + and 2R + are odd numbers. 90. a. a. 2N + 2R = 2(N + R). Since it has 2 as a factor, 2(N + R) is an even number. The sum of two even numbers is even. b. (2N + ) + (2R + ) = 2N + 2R + 2 = 2(N + R + ) The sum of two odd numbers is even. c. 2N + (2R + ) = 2N + 2R + = 2(N + R) + ) The number 2(N + R) is even. An even number plus is odd, so 2(N + R) + is odd (3) 3() n n 3 n 5 c. d. (2 + n)(3 + 5) = (2 + n)(3) + (2 + n)(5) = 2(3) + n(3) + 2(5) + n(5) = 6 + 3n n = 6 + 8n n n a n a a 2 3 (n + 2)(a + 3) = an + 2a + 3n n n 3 n 5 (a + b)(c + d) = a * c + a * d + b * c + b * d = ac + ad + bc + bd b. 7 b + c + d a ab ac ad 3(7) 9. Answers will vary. Possible answers are = 0, ()(2)(3)(4) = 24, 4, = 4, and 2 * 3 4 = 8. Prime Time Investigation 4

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