Solved Exercises on Least Common Multiple (LCM) in Grade 6

Master LCM: listing method, prime factorization, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Listing Method
Exercise 1
Find the LCM of 6 and 8 by listing the first several multiples of each number and identifying the least common multiple.
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers

Listing method:
  1. List multiples of first number: Write several multiples of the first number
  2. List multiples of second number: Write several multiples of the second number
  3. Identify common multiples: Find numbers that appear in both lists
  4. Select the least: Choose the smallest common multiple
Multiples of 6:
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
6 × 5 = 30
6 × 6 = 36
6 × 7 = 42
6 × 8 = 48
Multiples of 6: {6, 12, 18, 24, 30, 36, 42, 48, ...}
Multiples of 8:
8 × 1 = 8
8 × 2 = 16
8 × 3 = 24
8 × 4 = 32
8 × 5 = 40
8 × 6 = 48
8 × 7 = 56
8 × 8 = 64
Multiples of 8: {8, 16, 24, 32, 40, 48, 56, 64, ...}
Step 1: List first several multiples of 6

{6, 12, 18, 24, 30, 36, 42, 48, ...}

Step 2: List first several multiples of 8

{8, 16, 24, 32, 40, 48, 56, 64, ...}

Step 3: Identify common multiples

Common multiples: {24, 48, ...}

Step 4: Find the least common multiple

The smallest number in {24, 48, ...} is 24

Step 5: Verify the result

24 ÷ 6 = 4 ✓

24 ÷ 8 = 3 ✓

LCM(6,8) = 24
Final answer:

The LCM of 6 and 8 is 24.

Applied rules:

• Multiple identification: Find numbers divisible by both original numbers

• Common multiple principle: LCM must be a multiple of all original numbers

• Verification: Check that LCM is divisible by both numbers

2 Prime Factorization Method
Exercise 2
Find the LCM of 12 and 18 using prime factorization. Show your work step by step.
Definition:

Prime factorization method: Finding LCM by taking the highest power of all prime factors

Prime Factorization of 12:

12 = 2² × 3¹

12 ÷ 2 = 6

6 ÷ 2 = 3

3 ÷ 3 = 1

Prime Factorization of 18:

18 = 2¹ × 3²

18 ÷ 2 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

Step 1: Find prime factorization of 12

12 = 2² × 3¹

Step 2: Find prime factorization of 18

18 = 2¹ × 3²

Step 3: Identify all prime factors

All primes: 2 and 3

Step 4: Take highest power of each prime

For 2: max(2, 1) = 2

For 3: max(1, 2) = 2

Step 5: Calculate LCM

LCM = 2² × 3² = 4 × 9 = 36

Step 6: Verify the result

36 ÷ 12 = 3 ✓

36 ÷ 18 = 2 ✓

LCM(12,18) = 36
Final answer:

The LCM of 12 and 18 is 36.

Applied rules:

• Prime factorization: Express each number as product of primes

• Complete coverage: Include all primes from both numbers

• Highest power rule: Take maximum exponent for each prime

3 Three Numbers
Exercise 3
Find the LCM of 8, 12, and 18 using the prime factorization method. Explain your approach.
Definition:

LCM of multiple numbers: The smallest number that is a multiple of all given numbers

Prime Factorizations:

8 = 2³

12 = 2² × 3¹

18 = 2¹ × 3²

Step 1: Find prime factorization of 8

8 = 2³

Step 2: Find prime factorization of 12

12 = 2² × 3¹

Step 3: Find prime factorization of 18

18 = 2¹ × 3²

Step 4: Identify all prime factors in any number

All primes: 2 and 3

Step 5: Take highest power of each prime

For 2: max(3, 2, 1) = 3

For 3: max(0, 1, 2) = 2

Step 6: Calculate LCM

LCM = 2³ × 3² = 8 × 9 = 72

Step 7: Verify the result

72 ÷ 8 = 9 ✓

72 ÷ 12 = 6 ✓

72 ÷ 18 = 4 ✓

LCM(8,12,18) = 72
Final answer:

The LCM of 8, 12, and 18 is 72.

Applied rules:

• Extension principle: Same method applies to multiple numbers

• Complete coverage: Include all primes from ANY number

• Maximum exponent: Take largest power among all numbers

LCM Methods and Properties
LCM(a,b) × GCF(a,b) = a × b
Fundamental Relationship
Method 1
Listing
Find common multiples
Method 2
Prime Factorization
Highest power of all primes
Method 3
Using GCF
LCM = (a × b) ÷ GCF(a,b)
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers

Multiple: A number that can be divided evenly by another number

Common multiple: A multiple that two or more numbers share

Prime factorization: Expressing a number as a product of prime numbers

LCM Finding Process:
  1. Choose method: Decide between listing or prime factorization
  2. Find multiples: Either list multiples or find prime factorization
  3. Identify commonality: Find multiples shared by all numbers
  4. Select minimum: Choose the smallest common multiple
  5. Verify: Ensure the LCM is divisible by all original numbers
Tip 1: For small numbers (≤ 10), listing method is often faster.
Tip 2: For large numbers, prime factorization method is more efficient.
Tip 3: Always check your answer by dividing the LCM by each original number.
Tip 4: The LCM is never smaller than the largest of the original numbers.
Common errors: Not finding all multiples, taking lowest instead of highest powers, not including all prime factors.
Success habits: Working systematically, double-checking work, using divisibility rules.
Important Properties:

• LCM(a,b) ≥ max(a,b): LCM is never smaller than the largest number

• LCM(a,a) = a: Number with itself has LCM equal to itself

• LCM(a,1) = a: Any number with 1 has LCM equal to itself

• LCM relationship: LCM(a,b) × GCF(a,b) = a × b

• Coprime numbers: If GCF(a,b) = 1, then LCM(a,b) = a × b

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
Two buses leave the station at the same time. Bus A returns every 12 minutes and Bus B returns every 18 minutes. After how many minutes will both buses return to the station at the same time again?
Definition:

Real-world application: Using LCM to find when simultaneous events occur again

Finding LCM of 12 and 18:

Prime factorization method:

12 = 2² × 3¹

18 = 2¹ × 3²

Common primes: 2 and 3

Highest powers: 2² and 3²

LCM = 2² × 3² = 4 × 9 = 36

Step 1: Identify the problem type

Both buses return at different intervals

We need to find when they'll coincide again

Step 2: Recognize this is an LCM problem

Bus A returns every 12 minutes

Bus B returns every 18 minutes

Find LCM(12, 18) to find when both return together

Step 3: Find prime factorization of 12

12 = 2² × 3¹

Step 4: Find prime factorization of 18

18 = 2¹ × 3²

Step 5: Identify all prime factors

All primes: 2 and 3

Step 6: Take highest power of each prime

For 2: max(2, 1) = 2

For 3: max(1, 2) = 2

Step 7: Calculate LCM

LCM = 2² × 3² = 4 × 9 = 36

Step 8: Verify the result

36 ÷ 12 = 3 (Bus A returns 3 times)

36 ÷ 18 = 2 (Bus B returns 2 times)

Both buses return after 36 minutes
Final answer:

Both buses will return to the station at the same time after 36 minutes.

Applied rules:

• Scheduling problem: LCM finds when cyclical events align

• Simultaneous occurrence: LCM gives time for next coincidence

• Verification: Check that LCM is divisible by both intervals

5 Advanced Problem
Exercise 5
Find the LCM of 15, 20, and 25. Then determine the smallest number of identical gift bags that can be created using all items if you have 15, 20, and 25 of three different types of items.
Definition:

Advanced application: Using LCM for optimal distribution problems

Prime Factorizations:

15 = 3¹ × 5¹

20 = 2² × 5¹

25 = 5²

All primes: 2, 3, 5

Highest powers: 2², 3¹, 5²

LCM = 2² × 3¹ × 5² = 4 × 3 × 25 = 300

Step 1: Find prime factorization of 15

15 = 3¹ × 5¹

Step 2: Find prime factorization of 20

20 = 2² × 5¹

Step 3: Find prime factorization of 25

25 = 5²

Step 4: Identify all prime factors

All primes in any number: 2, 3, 5

Step 5: Take highest power of each prime

For 2: max(0, 2, 0) = 2

For 3: max(1, 0, 0) = 1

For 5: max(1, 1, 2) = 2

Step 6: Calculate LCM

LCM = 2² × 3¹ × 5² = 4 × 3 × 25 = 300

Step 7: Interpret for gift bags

LCM tells us the total number of items in all bags

But for the number of bags, we need GCF: GCF(15, 20, 25) = 5

So we can make 5 identical bags with 3, 4, 5 of each item respectively

LCM(15,20,25) = 300
Final answer:

The LCM of 15, 20, and 25 is 300. The smallest number of identical gift bags that can be created is 5 bags, each containing 3, 4, and 5 of each item type respectively.

Applied rules:

• Three-number LCM: Same method extends to multiple numbers

• Distribution application: LCM gives total for equal distribution

• Bag creation: GCF determines number of identical bags

Complete Summary: Least Common Multiple (LCM)
LCM(a,b) = min{m : a|m and b|m}
LCM Definition
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers

Multiple: A number that can be divided evenly by another number (b is a multiple of a if b ÷ a is a whole number)

Common multiple: A multiple that two or more numbers share

Coprime numbers: Two numbers whose GCF is 1 (they share no common factors except 1)

Complete methodology:
  1. Identify the numbers: Determine which numbers to find LCM for
  2. Choose method: Select listing or prime factorization based on number size
  3. Execute method: Either list multiples or find prime factorizations
  4. Find commonality: Identify multiples shared by all numbers
  5. Select minimum: Choose the smallest common multiple
  6. Verify result: Ensure LCM is divisible by all original numbers
Tip 1: For numbers under 10, listing method is usually faster.
Tip 2: For larger numbers, prime factorization method is more efficient.
Tip 3: Use the relationship LCM(a,b) × GCF(a,b) = a × b to verify results.
Tip 4: Remember that LCM is always at least as large as the largest number.
Key applications: Adding/subtracting fractions, scheduling problems, finding common periods, real-world timing scenarios.
Mathematical significance: Forms the basis for understanding divisibility and number relationships.
Essential Rules:

• Size requirement: LCM is never smaller than the largest number

• Upper bound: LCM is at most the product of all numbers

• Self-property: LCM(a, a) = a

• Unity property: LCM(a, 1) = a

• Prime property: If p is prime and p doesn't divide a, then LCM(p, a) = p × a

• Relationship: LCM(a, b) × GCF(a, b) = a × b

Questions & Answers

Question: When should I use the listing method versus the prime factorization method for finding LCM?

Answer: Choose your method based on the numbers:

Use listing method when:

  • Numbers are small (≤ 10)
  • You can easily list the first several multiples
  • Numbers have few small multiples
  • You're still learning the concept

Use prime factorization method when:

  • Numbers are large
  • Numbers have many factors
  • You need to find LCM of more than two numbers
  • You want to see the mathematical relationship clearly

Example: For LCM(4, 6), listing is quick: multiples of 4: {4, 8, 12, 16, ...}, multiples of 6: {6, 12, 18, ...}, LCM = 12.

For LCM(24, 36), prime factorization is more efficient: 24 = 2³ × 3¹, 36 = 2² × 3², LCM = 2³ × 3² = 72.

Both methods give the same answer - choose the one that's faster for your specific numbers!

Question: Can the LCM of two numbers ever be one of the original numbers? When does this happen?

Answer: Yes, the LCM can be one of the original numbers! This happens when one number is a multiple of the other.

For example:

  • LCM(6, 12) = 12 (because 12 is a multiple of 6)
  • LCM(8, 24) = 24 (because 24 is a multiple of 8)
  • LCM(5, 15) = 15 (because 15 is a multiple of 5)

In general, if a divides b (meaning b ÷ a is a whole number), then LCM(a, b) = b.

The most extreme case is LCM(a, a) = a - any number's LCM with itself is the number itself.

On the other hand, if two numbers are coprime (their GCF is 1), then LCM(a, b) = a × b.

Example: LCM(7, 10) = 70 because 7 and 10 are coprime (GCF(7, 10) = 1).

Question: How can I check if my LCM answer is correct? Is there a quick way to verify?

Answer: Yes, there are several ways to verify your LCM:

  1. Division check: Divide your LCM by each original number. The results should be whole numbers with no remainder.
  2. Multiple check: Make sure your LCM is actually a multiple of all original numbers.
  3. Relationship check: Use the formula LCM(a, b) × GCF(a, b) = a × b to verify.

Example: If you found LCM(12, 18) = 36:

  • 36 ÷ 12 = 3 ✓ (whole number)
  • 36 ÷ 18 = 2 ✓ (whole number)
  • Is it the smallest? Check multiples of 12: {12, 24, 36, ...} and 18: {18, 36, ...}, so 36 is indeed the smallest common multiple

You can also use the relationship: LCM(12, 18) × GCF(12, 18) = 36 × 6 = 216 = 12 × 18 ✓

The division check is the quickest verification method!