Q1: In a sports day event, students are lined up in rows of 6, 9, or 18. What is the smallest number of students that can be arranged in this way?
Sol: To determine the smallest number of students that can be arranged in rows of 6, 9, or 18, we need to calculate the Least Common Multiple (LCM) of the numbers 6, 9, and 18.
Prime factors:
- 6 = 2 × 3
- 9 = 3 × 3
- 18 = 2 × 3 × 3
Common factors and multiplication:
- Common factors = 2, 3 × 3
- Multiply all common factors: 2 × 3 × 3 = 18
Therefore, the smallest number of students is 18.
Q2: A baker has cookies that he wants to pack in boxes of 7, 14, or 21 cookies. What is the least number of cookies he needs to ensure there are no cookies left out?
Sol: To ensure no cookies are left out when packed in boxes of 7, 14, or 21, we find the LCM of these numbers.
Prime factors:
- 7 = 7
- 14 = 2 × 7
- 21 = 3 × 7
Common factors and multiplication:
- Common factors = 7
- Multiply the greatest number of times each prime factor appears in the factorization: 2 × 3 × 7 = 42
Therefore, the baker needs at least 42 cookies.
Q3: At a community picnic, the organizers want to divide the attendees into groups of 4, 6, or 8 evenly. What is the minimum number of attendees needed?
Sol: To find the minimum number of attendees that can be divided evenly into groups of 4, 6, or 8, we calculate the LCM of these numbers.
Prime factors:
- 4 = 2 × 2
- 6 = 2 × 3
- 8 = 2 × 2 × 2
Common factors and multiplication:
- Common factors = 2 × 2 × 2, 3
- Multiply these together: 2 × 2 × 2 × 3 = 24
Therefore, at least 24 attendees are needed.
Q4: A classroom has students who need to be arranged in rows of 10, 20, or 25 for a group photo. What is the least number of students that should be present?
Sol: To find the least number of students that can be arranged in rows of 10, 20, or 25, we look for the LCM.
Prime factors:
- 10 = 2 × 5
- 20 = 2 × 2 × 5
- 25 = 5 × 5
Common factors and multiplication:
- Common factors = 2 × 2, 5 × 5
- Multiply these together: 2 × 2 × 5 × 5 = 100
Therefore, at least 100 students should be present.
Q5: A musical concert is organized where the audience must be seated in sections of 15, 30, or 45. How many minimum seats should be available?
Sol: To find the minimum number of seats, calculate the LCM of 15, 30, and 45.
Prime factors:
- 15 = 3 × 5
- 30 = 2 × 3 × 5
- 45 = 3 × 3 × 5
Common factors and multiplication:
- Common factors = 2, 3 × 3, 5
- Multiply these together: 2 × 3 × 3 × 5 = 90
Therefore, at least 90 seats should be available. These solutions use the factorization method to find the LCM, ensuring that students can understand the process through basic multiplication of common prime factors.
Q6: At a dance competition, dancers are grouped into teams of 5, 10, or 15. What is the smallest team size that allows all configurations?
Sol: To find the smallest team size that can be divided into groups of 5, 10, or 15, we need to calculate the Least Common Multiple (LCM).
Prime factors:
- 5 = 5
- 10 = 2 × 5
- 15 = 3 × 5
Common factors and multiplication:
- The LCM needs the highest power of all primes present: 2, 3, 5
- Multiply these together: 2 × 3 × 5 = 30
Therefore, the smallest team size that allows all configurations is 30.
Q7: A gardener wants to plant tulips in rows that could evenly make up 3, 6, or 9 rows. What is the least number of tulips he needs?
Sol: The least number of tulips needed to divide evenly into rows of 3, 6, or 9 is found by determining the LCM of these numbers.
Prime factors:
- 3 = 3
- 6 = 2 × 3
- 9 = 3 × 3
Common factors and multiplication:
- Common factors = 3 × 3
- Multiply these together: 3 × 3 = 9
Therefore, the gardener needs at least 9 tulips.
Q8: What is the smallest number of candies that can be evenly divided among 9, 12, or 15 friends without any left over?
Sol: To find the least number of candies that can be evenly divided among 9, 12, or 15 friends, we need to calculate the Least Common Multiple (LCM) of these numbers.
Here’s how we find the LCM:
List the prime factors:
- 9 = 3 × 3
- 12 = 2 × 2 × 3
- 15 = 3 × 5
Choose the highest powers of all prime numbers involved:
- The highest power of 2 appearing in the factorization is 2 × 2.
- The highest power of 3 appearing is 3 × 3.
- The highest power of 5 is 5.
Multiply these together to find the LCM:
- LCM = (2 × 2) × (3 × 3) × 5 = 4 × 9 × 5 = 36 × 5 = 180
Therefore, the smallest number of candies is 180.
Q9: A farmer wants to pack his apples into bags of 5, 10, or 15. What is the minimum number of apples he needs so that no apples are left out of the bags?
Sol: To ensure no apples are left out when packing into bags of 5, 10, or 15, we need to find the LCM of these numbers:
List the prime factors:
- 5 = 5
- 10 = 2 × 5
- 15 = 3 × 5
Choose the highest powers of all prime numbers involved:
- The highest power of 2 is 2.
- The highest power of 3 is 3.
- The highest power of 5 is 5.
Multiply these together to find the LCM:
- LCM = 2 × 3 × 5 = 6 × 5 = 30
Therefore, the farmer needs a minimum of 30 apples.
Q10: During a school play, students need to be grouped into teams of either 8 or 12. What is the least number of students needed to form both types of teams without any student left out?
Sol: For teams of 8 and 12, the LCM is calculated as follows:
List the prime factors:
- 8 = 2 × 2 × 2
- 12 = 2 × 2 × 3
Choose the highest powers of all prime numbers involved:
- The highest power of 2 is 2 × 2 × 2.
- The highest power of 3 is 3.
Multiply these together to find the LCM:
- LCM = (2 × 2 × 2) × 3 = 8 × 3 = 24
Therefore, at least 24 students are needed.
58 videos|125 docs|40 tests
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1. What are factors and how do you find them for a given number? | ![]() |
2. What are multiples and how can I calculate them? | ![]() |
3. What is the difference between factors and multiples? | ![]() |
4. Can you give examples of finding factors and multiples for the number 12? | ![]() |
5. How do prime numbers relate to factors and multiples? | ![]() |