Q1: Ravi buys pencils in packs of 6. He wants to buy enough packs so that he has exactly 36 pencils. How many packs should he buy?
Solution:
Each pack has 6 pencils.
Total pencils needed = 36
Number of packs = 36 ÷ 6 = 6 packs
Answer: 6 packs
Q2: A clothing store packs shirts into bundles of 4, 8, or 12. What is the smallest number of shirts the store should have to make up the bundles without any left?
Sol: To ensure no shirts are left over when packing into bundles of 4, 8, or 12, find the LCM.
Prime factors:
- 4 = 2 × 2
- 8 = 2 × 2 × 2
- 12 = 2 × 2 × 3
Common factors and multiplication:
- Common factors = 2 × 2 × 2, 3
- Multiply these together: 2 × 2 × 2 × 3 = 24
Therefore, the store should have at least 24 shirts.
Q3: A rectangular garden has an area of 24 square meters. Its length and width must be whole numbers. List all possible pairs of length and width.
Solution:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Possible pairs: (1, 24), (2, 12), (3, 8), (4, 6)
Answer: (1, 24), (2, 12), (3, 8), (4, 6)
Q4: A bus comes every 15 minutes and a train comes every 20 minutes. If both come together at 8:00 AM, when will they next come together?
Solution:
Find LCM of 15 and 20
Multiples of 15: 15, 30, 45, 60
Multiples of 20: 20, 40, 60
LCM = 60 minutes → 1 hour
Answer: Next at 9:00 AM
Q5: During a clean-up event, trash bags were filled in groups of 3, 6, or 9 without leaving any trash out. What is the minimum number of trash bags needed?
Sol: The minimum number of trash bags needed to organize trash into groups of 3, 6, or 9 is the LCM of these numbers.
Prime factors:
- 3 = 3
- 6 = 2 × 3
- 9 = 3 × 3
Common factors and multiplication:
- The highest power of 3 present is 3 × 3
- Multiply this with 2: 2 × 3 × 3 = 18
Therefore, at least 18 trash bags are needed for the clean-up event.
These solutions methodically determine the LCM using prime factorization, ensuring that all common factors are considered and multiplied to find the required minimum number. This approach helps clearly explain how to solve these types of problems in a structured way.
Q6: Sita has 18 apples and wants to put them into baskets so that each basket has the same number of apples. How many apples can she put in each basket?
Solution:
Factors of 18: 1, 2, 3, 6, 9, 18
She can have 1, 2, 3, 6, 9, or 18 apples per basket.
Answer: 1, 2, 3, 6, 9, or 18 apples per basket
Q7: A light flashes every 12 seconds and another light flashes every 18 seconds. If they flash together at 6:00 PM, when will they flash together again?
Solution:
LCM of 12 and 18
Multiples of 12: 12, 24, 36, 48, 60, 72…
Multiples of 18: 18, 36, 54, 72…
LCM = 36 seconds
Answer: They will flash together again after 36 seconds
Q8: For an art project, beads are needed in sets of 5, 10, or 20. What is the minimum number of beads required?
Sol: The smallest number of beads required to form sets of 5, 10, or 20 evenly is found by calculating the LCM.
Prime factors:
- 5 = 5
- 10 = 2 × 5
- 20 = 2 × 2 × 5
Common factors and multiplication:
- The highest power of primes: 2 × 2, 5
- Multiply together: 2 × 2 × 5 = 20
Therefore, a minimum of 20 beads are needed for the art project.
Q9: A teacher has 30 students. She wants to arrange them in rows so that each row has the same number of students. How many ways can she arrange them?
Solution:
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
So, possible arrangements: 1×30, 2×15, 3×10, 5×6, 6×5, 10×3, 15×2, 30×1
Answer: 8 possible arrangements
Q10: A rectangle has an area of 32 sq. meters. Find all possible pairs of whole-number length and width.
Solution:
Factors of 32: 1, 2, 4, 8, 16, 32
Possible pairs: (1,32), (2,16), (4,8), (8,4), (16,2), (32,1)
Answer: (1,32), (2,16), (4,8), (8,4), (16,2), (32,1)
Q11: In a school laboratory, specimens need to be placed in containers holding 4, 8, or 16 specimens each. How many minimum specimens are required?
Sol: The smallest number of specimens required to fit into containers of 4, 8, or 16 evenly is determined by the LCM.
Prime factors:
- 4 = 2 × 2
- 8 = 2 × 2 × 2
- 16 = 2 × 2 × 2 × 2
Common factors and multiplication:
- The highest power of 2 present: 2 × 2 × 2 × 2
- Multiply this: 2 × 2 × 2 × 2 = 16
Therefore, at least 16 specimens are required in the laboratory.
Q12: Sita has 24 apples. She wants to put them in baskets with equal numbers. How many apples can each basket have?
Solution:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Answer: 1, 2, 3, 4, 6, 8, 12, or 24 apples
Q13: A gardener waters plants every 6 days. Another gardener waters plants every 9 days. Both water plants today. When will they water plants together next?
Solution:
LCM of 6 and 9 = 18 days
After 18 days, they will water together.
Q14: A construction site has workers who must be grouped into teams of 5, 10, or 15 for different tasks. What is the minimum number of workers needed?
Sol: To find the smallest number of workers that can be grouped into teams of 5, 10, or 15 without any worker left out, we need to calculate the LCM.
Prime factors:
- 5 = 5
- 10 = 2 × 5
- 15 = 3 × 5
Common factors and multiplication:
- The LCM includes the highest power of all primes: 2, 3, 5
- Multiply these together: 2 × 3 × 5 = 30
Therefore, at least 30 workers are needed at the construction site.
Q15: A teacher has 36 students. She wants to arrange them in equal rows. How many ways can she do it?
Solution:
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Possible rows: 1×36, 2×18, 3×12, 4×9, 6×6, 9×4, 12×3, 18×2, 36×1
Therefore, there are 9 ways
