This question tests correcting a misconception about LCM, not always the product. LCM is smallest common multiple, like LCM(6,8)=24, not 48. Student said 48, a multiple but not least; correct LCM=24 from multiples. Statement B is correct: incorrect, LCM is 24. Errors: thinking product always LCM (only if coprime), or LCM smaller than numbers. Find LCM by multiples list or primes: 6=2*3, $8=2^3$, $LCM=2^3$*3=24. Applications: timing like bus schedules.

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 $54=23^3$ and $72=2^3$$*3^2$, the $GCF=2^1$$3^2$=29=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 $54=23^2$ (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.

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=25, $12=2^2$3, $LCM=2^2$35=435=60. Applications include cycle alignment; mistakes include not picking least or listing errors.

This question tests finding LCM of 9 and 12, the smallest positive multiple of both. LCM is smallest both divide into, like LCM(6,8)=24 from multiples. For 9 and 12, multiples of 9={9,18,27,36,...}, 12={12,24,36,...}, common start at 36, so LCM=36. The correct answer is 36, choice C. Errors include picking GCF 3, or a larger multiple like 108, or arithmetic like 9+12=21. To find LCM: list multiples, find common, choose least; or prime factors $9=3^2$, $12=2^2$*3, $LCM=2^2$$*3^2$=36. Applications: cycles coinciding, like events every 9 and 12 days.

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.

This question tests using the distributive property to factor a sum like 36+8 using the GCF, where you factor out the greatest common factor and ensure the quotients inside are coprime, meaning they have no common factor greater than 1. The GCF is the largest number dividing both, like GCF(36,48)=12 from common factors {1,2,3,4,6,12}; LCM is the smallest both divide into, like LCM(6,8)=24. For example, to factor 36+8, GCF(36,8)=4, so 4(36/4 + 8/4)=4(9+2), and 9 and 2 are coprime since their factors are {1,3,9} and {1,2} with only 1 in common. The correct expression is 4(9+2), as it's using the GCF and fully reduced. A common error is factoring with a smaller number like 2(18+4), where 18 and 4 share a factor of 2, or using a non-factor like 8(4+1) but 8 doesn't divide 36 evenly since 36/8=4.5. To factor using GCF: (1) find GCF(a,b), (2) divide both by GCF, (3) write GCF(quotient1 + quotient2), (4) verify quotients are coprime. Applications include simplifying expressions; mistakes include not using the greatest factor, leaving common factors inside, or arithmetic errors like wrong division.

This question tests finding the LCM, the least common multiple or smallest number both divide into, like when events coincide again. LCM is the smallest number both divide into, for example, LCM(6,8) from multiples of 6={6,12,18,24,...} and 8={8,16,24,...}, common={24,48,...}, least=24. For this timer problem, list multiples to find the first common one at 24 seconds. The correct answer is 24, choice B, as that's when they flash together next. Errors include picking the product 48, which is a multiple but not the least, or a common factor like 2 instead. To find LCM, list multiples of each, find common multiples, and choose the smallest. Prime factorization helps: for LCM take highest powers of all primes; applications are scheduling like these lights.

This question tests GCF of 45 and 30 for making equal stations with all cones. GCF is largest dividing both, like GCF(36,48)=12. Factors of 45={1,3,5,9,15,45}, 30={1,2,3,5,6,10,15,30}, common={1,3,5,15}, greatest=15. Correct is 15 stations, choice C, with 45/15=3 orange and 30/15=2 yellow each. Error: picking smaller like 5, not greatest, or LCM 90 too big. Find GCF by listing factors, common, largest; or primes $45=3^2$5, 30=235, GCF=35=15. Use for equal divisions like here.

This question tests identifying GCF from given factor lists of 24 and 60. GCF is largest common factor, like from lists for 36 and 48 yielding 12. Given lists: common={1,2,3,4,6,12}, so greatest=12. Correct is 12, choice C. Error: picking smaller like 6, not greatest, or non-common like 8. Find GCF by comparing lists, pick largest common. Primes: $24=2^3$*3, $60=2^2$35, $GCF=2^2$*3=12; used for divisions.

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.