Find GCF and LCM - 6th Grade Math

Two bells ring on a schedule. One rings every 6 minutes and the other rings every 8 minutes. If they ring together at 12:00, after how many minutes will they ring together again? (Find $\text{LCM}(6,8)$ .)
Answer: 24 minutes This question tests finding the LCM, the least common multiple, the smallest number that is a multiple of both 6 and 8, useful for when events like bell rings coincide again. The LCM is the smallest number both divide into, for example, LCM(6,8): multiples of 6 are {6,12,18,24,...}, multiples of 8 are {8,16,24,...}, common multiples are {24,48,...}, least is 24; for GCF, it's the largest dividing both, like GCF(36,48)=12. Using the distributive property with GCF, for 36+8 with GCF=4, it's 4(9+2) where 9 and 2 are coprime. Here, they ring together again after LCM(6,8)=24 minutes, as 24 is the smallest common multiple. A common error is choosing a larger multiple like 48, or confusing with GCF which is 2, or adding instead of finding multiples. To find LCM, list multiples of each, find common ones, and choose the smallest; or use prime factorization: 6=2*3, 8= $2^3$ , LCM= $2^3$ *3=8*3=24. Applications include scheduling, like when cycles align; mistakes include picking non-multiples like 14 or arithmetic errors in listing.
Which choice correctly finds the greatest common factor using prime factorization? $72=2^3\cdot 3^2$ and $90=2\cdot 3^2\cdot 5$
Answer: $\text{GCF}=2\cdot 3^2=18$ This question tests finding the GCF using prime factorization, where GCF takes the lowest powers of common primes for 72= $2^3$ * $3^2$ and 90=2* $3^2$ *5. The GCF is largest dividing both, for example, common primes 2 and 3, lowest powers $2^1$ * $3^2$ =18. In contrast, LCM uses highest powers, like $2^3$ * $3^2$ *5=360. For example, GCF(72,90): min( $2^3,2^1$ )= $2^1$ , min( $3^2,3^2$ )= $3^2$ , no 5 in 72, so 2*9=18. The correct choice is 2* $3^2$ =18. A common mistake is using highest powers like 72 or including extra like 360 or 45. To find GCF with primes: factor both, take lowest powers of commons; this verifies factorizations and applies to simplifying fractions or grouping.
Find the greatest common factor of 54 and 72 by using prime factorization. Which choice correctly gives the prime factorizations and the GCF?
Answer: $54=2\cdot 3^3$ , $72=2^3\cdot 3^2$ , so $\text{GCF}=2\cdot 3^2=18$ This question tests finding the GCF of 54 and 72 using prime factorization, where GCF is the product of the lowest powers of common primes, helping in simplifying fractions or dividing evenly. The GCF is the largest number dividing both, for example, GCF(36,48)=12; LCM uses highest powers, like LCM(6,8)=24 from $2^3$ 3. For 54=2 $3^3$ and 72= $2^3$ * $3^2$ , the GCF= $2^1$ * $3^2$ =2*9=18, as it takes the minimum exponents. The correct choice is the one with accurate factorizations and GCF=18. A common error is using highest exponents like for LCM, getting $2^3$ * $3^3$ =216, or wrong factorization like 54=2* $3^2$ (which is 18, not 54). To find GCF with primes: factor each number, take common primes with lowest powers, multiply; for LCM, use highest powers. Applications include reducing ratios; mistakes include incorrect exponents, confusing GCF with LCM, or arithmetic errors in multiplication.
A school band practices every 10 days and the choir practices every 12 days. If both groups practice today, in how many days will they practice on the same day again (the LCM of 10 and 12)?
Answer: 60 This question tests LCM of 10 and 12 for when practices coincide. LCM is smallest both divide into, like LCM(6,8)=24. Multiples 10={10,20,30,40,50,60,...}, 12={12,24,36,48,60,...}, least common=60. Correct is 60 days, choice C. Error: picking GCF 2, or product 120 (not least), or 10+12=22. Find LCM by multiples or primes: 10=2*5, 12= $2^2$ *3, LCM= $2^2$ *3*5=60. Use for cycle alignments like schedules.
A student says: " $\text{LCM}(6,8)=48$ because $6\times 8=48$ ." Which value is actually $\text{LCM}(6,8)$ (the smallest common multiple)?
Answer: 24 This question tests identifying the actual LCM of 6 and 8, correcting the student's mistake of using the product instead of the least common multiple. The LCM is the smallest number both divide into, like LCM(6,8)=24 from multiples; GCF is the largest dividing both, like GCF(36,48)=12. For example, multiples of 6: {6,12,18,24,...}, of 8: {8,16,24,...}, common: {24,48,...}, least=24, not 48 which is larger. The actual LCM is 24, the smallest common multiple. A common error is multiplying like 6*8=48, or choosing non-multiples like 12 (not of 8) or 16 (not of 6). To find LCM: list multiples, find common, choose smallest; or primes: 6=2*3, 8= $2^3$ , LCM= $2^3$ *3=24. Applications include timing; mistakes include using product or not least.
A student is finding $\text{LCM}(9,12)$ . Which answer is the least common multiple of 9 and 12?
Answer: 36 This question tests finding the LCM of 9 and 12, emphasizing it's the least common multiple, the smallest number divisible by both, useful for aligning cycles or adding fractions. The LCM is the smallest number both divide into, like LCM(6,8)=24; GCF is the largest dividing both, like GCF(36,48)=12. For example, multiples of 9: {9,18,27,36,...}, multiples of 12: {12,24,36,48,...}, common: {36,72,...}, least=36. The least common multiple is 36, as smaller options like 18 divides 9 but not 12, and 24 divides 12 but not 9. A common error is picking a larger multiple like 108, or confusing with GCF=3, or choosing a non-multiple. To find LCM: list multiples, find common, choose smallest; or primes: 9= $3^2$ , 12= $2^2$ *3, LCM= $2^2$ * $3^2$ =4*9=36. Applications include least time for events to sync; mistakes include not selecting the least, or arithmetic errors in multiples.
A teacher has 36 red pencils and 48 blue pencils. She wants to make identical gift bags with no pencils left over. What is the greatest number of gift bags she can make (the GCF of 36 and 48)?
Answer: 12 This question tests finding the GCF, which is the greatest common factor or the largest number that divides both without remainder, and relates to making equal groups like gift bags using all items. The GCF is the largest number that divides both, for example, GCF(36,48) from factors of 36={1,2,3,4,6,9,12,18,36} and 48={1,2,3,4,6,8,12,16,24,48}, common={1,2,3,4,6,12}, so greatest=12. For this problem, list factors of 36 and 48 to find common ones, identifying 12 as the greatest, meaning the most bags is 12, with 36/12=3 red and 48/12=4 blue per bag. The correct answer is 12, choice B, as it allows identical bags with no leftovers using the maximum number. A common error is picking a smaller common factor like 6, which works but isn't the greatest, or confusing with LCM like 144 which is too big. To find GCF, list factors of both numbers, identify all common ones, and choose the largest. Applications include dividing items into equal groups, like the largest number of bags here.
Jake needs to arrange chairs in rows for two different events. For Event A, he has 36 chairs, and for Event B, he has 60 chairs. He wants to use the same number of chairs per row for both events, with no chairs left over. What is the greatest number of chairs he can put in each row?
Answer: 12 chairs per row for both events We need the GCF of 36 and 60. Since 36 = 2² × 3² and 60 = 2² × 3 × 5, the GCF is 2² × 3 = 12. This means 12 chairs per row is the maximum possible. Choice A (6 chairs) is a common factor but not the greatest, Choice C (18 chairs) doesn't divide 60 evenly (60 ÷ 18 = 3.33...), and Choice D (4 chairs) is a common factor but not the greatest.
A rectangular bulletin board is 45 inches wide and 60 inches tall. The art club wants to cover it with identical square photos, with no cutting and no gaps. What is the side length (in inches) of the largest square photo they can use? (This is $\text{GCF}(45,60)$ .)
Answer: 15 This question tests finding the GCF of 45 and 60, which gives the largest square side that divides both dimensions evenly for covering without gaps or cuts. The GCF is the largest number dividing both, like GCF(36,48)=12; LCM is the smallest both divide into, like LCM(6,8)=24. For example, factors of 45: {1,3,5,9,15,45}, factors of 60: {1,2,3,4,5,6,10,12,15,20,30,60}, common: {1,3,5,15}, greatest=15. The side length of the largest square is 15 inches, fitting 45/15=3 across and 60/15=4 up. A common error is choosing a smaller factor like 5 or 10, or confusing with LCM=180. To find GCF: list factors, identify common, choose largest; or primes: 45= $3^2$ *5, 60= $2^2$ *3*5, GCF=3*5=15. Applications include tiling or grouping; mistakes include not picking greatest, or errors in factor listing.
A student says: "The GCF of 36 and 48 is 6 because 6 is a common factor." Which statement is correct?
Answer: The student is incorrect; the GCF must be the largest common factor, which is 12. This question tests understanding GCF as the greatest common factor, not just any, addressing a student's misconception. GCF is the largest dividing both, like GCF(36,48)=12 from common factors {1,2,3,4,6,12}. The student picked 6, a common factor, but not the greatest, so incorrect; the largest is 12. The correct statement is B, that the student is incorrect and GCF is 12. Errors include thinking any common factor is GCF, or confusing with smallest like 1, or believing 6 is only one. To find GCF, list factors, find common, pick largest; prime factors use lowest powers. This clarifies definitions for applications like grouping.
Three alarm clocks ring at different intervals. Clock A rings every 6 minutes, Clock B rings every 9 minutes, and Clock C rings every 4 minutes. If all three clocks ring together at 3:00 PM, at what time will all three ring together again?
Answer: 3:36 PM When you see a problem about events happening at regular intervals and need to find when they'll occur together again, you're looking for the Least Common Multiple (LCM). This tells you the smallest amount of time that passes before all cycles repeat simultaneously. To find when all three clocks ring together again, you need the LCM of their intervals: 6 minutes, 9 minutes, and 4 minutes. Start by finding the prime factorization of each number: - 6 = 2 × 3 - 9 = 3² - 4 = 2² For the LCM, take the highest power of each prime factor that appears: 2² × 3² = 4 × 9 = 36. So all three clocks will ring together every 36 minutes. Since they all rang at 3:00 PM, add 36 minutes: 3:00 PM + 36 minutes = 3:36 PM. Looking at the wrong answers: Choice A (3:18 PM) represents 18 minutes after 3:00, which is the LCM of just clocks A and C (6 and 4). Choice B (4:12 PM) is 72 minutes later, which would be when they ring together for the second time after 3:00. Choice C (3:54 PM) doesn't correspond to any meaningful LCM calculation and may result from adding the intervals incorrectly. Study tip: When solving "meeting again" problems, always find the LCM of all the given intervals. Don't confuse this with finding when just two of the items will coincide - you need all of them together.
Use the distributive property to factor out the greatest common factor from $63+27$ . Which expression is correct and fully reduced?
Answer: $9(7+3)$ This question tests using the distributive property to factor out the GCF from 63+27, ensuring the expression is fully reduced with coprime terms inside. The distributive factors as GCF*(a/GCF + b/GCF) with coprime quotients, for example, GCF(63,27)=9, 63/9=7, 27/9=3, so 9(7+3) where 7 and 3 are coprime. In contrast, using 3 gives 3(21+9), but not greatest, and 21+9 share 3. For example, factor 63+27: GCF=9, quotients 7+3, coprime. The correct expression is 9(7+3). A common mistake is using non-GCF like 27 with fractions or 6 with decimals. To factor: find GCF, divide, write expression, verify coprime; this aids in simplifying algebraic expressions or sums.
Two students start jogging at the same time. One completes a lap every 10 minutes and the other completes a lap every 12 minutes. If they start together, after how many minutes will they both be at the starting line together again? (Find $\text{LCM}(10,12)$ .)
Answer: 60 This question tests finding the LCM of 10 and 12, the smallest time when joggers meet again at the start, aligning their lap cycles. The LCM is the smallest number both divide into, like LCM(6,8)=24; GCF is the largest dividing both, like GCF(36,48)=12. For example, multiples of 10: {10,20,30,40,50,60,...}, of 12: {12,24,36,48,60,...}, common: {60,120,...}, least=60. They meet after 60 minutes, as smaller like 20 isn't a multiple of 12. A common error is choosing GCF=2, or a non-multiple like 22, or product 120 which isn't least. To find LCM: list multiples, find common, choose smallest; or primes: 10=2*5, 12= $2^2$ *3, LCM= $2^2$ *3*5=4*3*5=60. Applications include cycle alignment; mistakes include not picking least or listing errors.
A school orders 24 chocolate snacks and 30 granola snacks. They want to pack them into identical bags with the same number of each type in every bag and no snacks left over. What is the greatest number of bags they can make? (Find $\text{GCF}(24,30)$ .)
Answer: 6 This question tests finding the GCF, the greatest common factor dividing 24 and 30, for the greatest number of identical snack bags with no leftovers. The GCF is largest dividing both, for example, factors of 24:1,2,3,4,6,8,12,24; 30:1,2,3,5,6,10,15,30; common 1,2,3,6; greatest 6. In contrast, LCM is for smallest multiple, but here GCF for bags. For example, GCF(24,30): list factors, common up to 6. The correct answer is 6 bags, with 24/6=4 chocolate and 30/6=5 granola each. A common mistake is smaller like 3 or unrelated like 12 (doesn't divide 30). To find GCF: list factors, pick largest common; or prime: 24= $2^3$ *3, 30=2*3*5, GCF=2*3=6; applies to packing or dividing resources evenly.
Two friends jog on a circular track. Sarah completes one lap every 8 minutes, and David completes one lap every 10 minutes. They start together at the starting line. How many laps will Sarah have completed when they meet at the starting line for the third time?
Answer: 15 laps First find when they meet: LCM(8, 10) = 40 minutes. They meet at the start at 0, 40, 80, and 120 minutes. The third meeting is at 120 minutes. In 120 minutes, Sarah completes 120 ÷ 8 = 15 laps. Choice B (12 laps) is what she completes at the second meeting (80 minutes), Choice C (30 laps) doubles the correct answer, and Choice D (10 laps) is David's lap count at the third meeting.
Two buses leave the station at the same time. Bus A returns to the station every 8 minutes, and Bus B returns every 12 minutes. After how many minutes will both buses be at the station together again for the second time?
Answer: After 48 minutes from the start We need the LCM of 8 and 12. Since 8 = 2³ and 12 = 2² × 3, the LCM is 2³ × 3 = 24. The buses meet at the station together at 0 minutes (start), 24 minutes (first time together again), and 48 minutes (second time together again). Choice A gives the first time they meet again, Choice C is 1.5 × 24 (not a multiple), and Choice D is 4 × 24 (fourth time together).
Maria has 48 stickers and 72 pencils to divide equally among some students. She wants each student to receive the same number of stickers and the same number of pencils, with no items left over. What is the maximum number of students that can receive items, and how many stickers will each student get?
Answer: 24 students, each getting 2 stickers To find the maximum number of students, we need the GCF of 48 and 72. Using prime factorization: 48 = 2⁴ × 3 and 72 = 2³ × 3². The GCF is 2³ × 3 = 24. So 24 students can receive items, and each gets 48 ÷ 24 = 2 stickers. Choice B uses 12 (a factor but not the greatest), Choice C uses 8 (a factor but not the greatest), and Choice D uses 6 (a factor but not the greatest).
A teacher has 36 red markers and 48 blue markers. She wants to make identical gift bags with no markers left over. What is the greatest number of gift bags she can make? (This is the GCF of 36 and 48.)
Answer: 12 This question tests finding the GCF, which is the greatest common factor, the largest number that divides both 36 and 48 without a remainder, useful for dividing items into equal groups like gift bags. The GCF is the largest number that divides both, for example, GCF(36,48): factors of 36 are {1,2,3,4,6,9,12,18,36}, factors of 48 are {1,2,3,4,6,8,12,16,24,48}, common factors are {1,2,3,4,6,12}, so greatest is 12. For LCM, it's the smallest number both divide into, like LCM(6,8): multiples of 6 are {6,12,18,24,...}, multiples of 8 are {8,16,24,...}, common multiples are {24,48,...}, least is 24. In this case, the greatest number of gift bags is the GCF(36,48)=12, so she can make 12 bags, each with 36/12=3 red and 48/12=4 blue markers. A common error is choosing a smaller common factor like 6, or confusing with LCM which is 144, or picking the product 36*48. To find GCF, list factors of both numbers, identify the common ones, and choose the largest; alternatively, use prime factorization: 36= $2^2$ * $3^2$ , 48= $2^4$ *3, GCF= $2^2$ *3=4*3=12. Applications include dividing resources equally, like the largest group size for markers; mistakes include not selecting the greatest factor or arithmetic errors in division.
Two students are making equal-length jump-rope pieces from two ropes: one is 45 ft and the other is 60 ft. They want the longest possible piece length that divides both exactly. What is $\text{GCF}(45,60)$ ?
Answer: 15 This question tests finding the GCF, which is the greatest common factor or the largest number that divides both 45 and 60 without remainder, representing the longest piece length. The GCF is the largest number dividing both, for example, GCF(45,60): factors of 45 are 1,3,5,9,15,45 and of 60 are 1,2,3,4,5,6,10,12,15,20,30,60, with common 1,3,5,15, greatest 15. In contrast, LCM would be for smallest common multiple, but here it's GCF for dividing lengths. For example, GCF(45,60) lists factors, common are 1,3,5,15, greatest=15. The correct answer is 15. A common mistake is choosing smaller like 5 or 10, or largest like 30 which doesn't divide 45. To find GCF: list factors, pick largest common; or prime: 45= $3^2$ *5, 60= $2^2$ *3*5, GCF=3*5=15; applies to dividing into equal parts like ropes or tiles.
A science class observes a flashing light every 9 seconds and a different flashing light every 12 seconds. If they flash together at time 0, when is the next time they flash together? (Find $\text{LCM}(9,12)$ .)
Answer: 36 seconds This question tests finding the LCM, which is the least common multiple or smallest multiple of both 9 and 12, representing the next time lights flash together. The LCM is the smallest both divide into, for example, LCM(9,12): multiples of 9 are 9,18,27,36,... and of 12 are 12,24,36,..., common 36,72,... least 36. In contrast, GCF is largest dividing both, but here LCM for timing. For example, LCM(9,12): list multiples, first common=36. The correct answer is 36 seconds. A common mistake is smaller common like 18 (not multiple of 12) or larger like 108. To find LCM: list multiples, pick smallest common; or prime: 9= $3^2$ , 12= $2^2$ *3, LCM= $2^2$ * $3^2$ =36; applies to events coinciding like lights or schedules.

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