Word Problem: Factors and Multiples - 2 | Mathematics for Class 5 PDF Download

Q1: A teacher wants to distribute school supplies in packs of 2, 4, or 8 equally. What is the least number of items she must have?

Sol: The smallest number of items that can be evenly distributed into packs of 2, 4, or 8 is the LCM of these numbers.

Prime factors:

• 2 = 2
• 4 = 2 × 2
• 8 = 2 × 2 × 2

Common factors and multiplication:

• The highest power of 2 present is 2 × 2 × 2
• Multiply this together: 2 × 2 × 2 = 8

Therefore, the teacher must have at least 8 items.

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: 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.

Q4: In a library, books are arranged on shelves in stacks of 7, 14, or 21. What is the least number of books needed for this arrangement?

Sol: To determine the least number of books needed to evenly stack them in groups of 7, 14, or 21, we calculate the Least Common Multiple (LCM).
Prime factors:

• 7 = 7
• 14 = 2 × 7
• 21 = 3 × 7

Common factors and multiplication:

• The LCM needs the highest power of all primes present: 2, 3, 7
• Multiply these together: 2 × 3 × 7 = 42

Therefore, the library needs at least 42 books to arrange them on shelves without any left over.

Q5: 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.

Q6: A sports league wants to form teams of 11, 22, or 33 players. What is the least number of players needed to form all types of teams evenly?

Sol: To find the least number of players that can be evenly divided into teams of 11, 22, or 33, we calculate the LCM.
Prime factors:

• 11 = 11
• 22 = 2 × 11
• 33 = 3 × 11

Common factors and multiplication:

• The highest power of all primes present: 2, 3, 11
• Multiply these together: 2 × 3 × 11 = 66

Therefore, at least 66 players are needed in the sports league.

Q7: 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.

Q8: A tour group is divided into smaller groups of 3, 6, or 12. What is the smallest number of tourists that allows such a division?

Sol: The minimum number of tourists needed for division into groups of 3, 6, or 12 is the LCM of these numbers.
Prime factors:

• 3 = 3
• 6 = 2 × 3
• 12 = 2 × 2 × 3

Common factors and multiplication:

• The highest power of primes: 2 × 2, 3
• Multiply these together: 2 × 2 × 3 = 12

Therefore, at least 12 tourists are required for the tour group division.

Q9: 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.

Q10: For a baking class, ingredients are packaged in quantities that can make batches of 2, 4, or 8 recipes. What is the least amount of each ingredient needed?

Sol: The smallest amount of ingredients required to make batches of 2, 4, or 8 recipes evenly is found by determining the LCM of these numbers.
Prime factors:

• 2 = 2
• 4 = 2 × 2
• 8 = 2 × 2 × 2

Common factors and multiplication:

• The highest power of 2 present: 2 × 2 × 2
• Multiply this together: 2 × 2 × 2 = 8