Solved Exercises on Least Common Multiple (LCM) in Pre-algebra

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

Solution: Exercises 1 to 3
1 Listing Multiples Method
Exercise 1
Find the LCM of 6 and 8 using the listing multiples method.
Definition:

Least Common Multiple (LCM): The smallest positive integer that is divisible by two or more numbers.

Listing Multiples Method: Find multiples of each number and identify the smallest common multiple.

Listing Multiples Method:
  1. List multiples of the first number
  2. List multiples of the second number
  3. Identify common multiples
  4. Select the smallest common multiple
Numbers
6 and 8
Multiples
6: 6,12,18,24... 8: 8,16,24...
LCM
24
Step 1: List multiples of 6

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

Step 2: List multiples of 8

8, 16, 24, 32, 40, 48, 56...

Step 3: Identify common multiples

Common multiples: 24, 48, 72...

Step 4: Select the smallest

The smallest common multiple is 24

LCM(6, 8) = 24
Final answer:

The LCM of 6 and 8 is 24.

Applied rules:

• Multiples: Find numbers that are divisible by each original number

• Commonality: Look for multiples shared by all numbers

• Least Value: Select the smallest common multiple

2 Prime Factorization Method
Exercise 2
Find the LCM of 12 and 18 using the prime factorization method.
Definition:

Prime Factorization Method: Find prime factorization of each number and multiply all prime factors with highest exponents.

Numbers
12 and 18
Prime Factorizations
2²×3 and 2×3²
LCM
2²×3² = 36
Step 1: Find prime factorization of 12

12 = 2² × 3 = 2 × 2 × 3

Step 2: Find prime factorization of 18

18 = 2 × 3² = 2 × 3 × 3

Step 3: Identify all prime factors

Prime factors present: 2 and 3

Highest exponent of 2: max(2,1) = 2

Highest exponent of 3: max(1,2) = 2

Step 4: Multiply all factors with highest exponents

LCM = 2² × 3² = 4 × 9 = 36

Step 5: Verification

36 ÷ 12 = 3 (whole number)

36 ÷ 18 = 2 (whole number)

✓ Both divisions result in whole numbers

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

• All Primes: Include every prime that appears in any factorization

• Highest Exponents: Use maximum exponent for each prime

3 Three Numbers LCM
Exercise 3
Find the LCM of 10, 15, and 25 using the prime factorization method.
Definition:

LCM of Multiple Numbers: The smallest positive integer that is divisible by all given numbers.

Numbers
10, 15, 25
Prime Factorizations
2×5, 3×5, 5²
LCM
2×3×5² = 150
Step 1: Find prime factorization of 10

10 = 2 × 5

Step 2: Find prime factorization of 15

15 = 3 × 5

Step 3: Find prime factorization of 25

25 = 5²

Step 4: Identify all prime factors

Prime factors present: 2, 3, and 5

Highest exponent of 2: max(1,0,0) = 1

Highest exponent of 3: max(0,1,0) = 1

Highest exponent of 5: max(1,1,2) = 2

Step 5: Multiply all factors with highest exponents

LCM = 2¹ × 3¹ × 5² = 2 × 3 × 25 = 150

Step 6: Verification

150 ÷ 10 = 15 (whole number)

150 ÷ 15 = 10 (whole number)

150 ÷ 25 = 6 (whole number)

✓ All divisions result in whole numbers

LCM(10, 15, 25) = 150
Final answer:

The LCM of 10, 15, and 25 is 150.

Applied rules:

• All Numbers: Include primes from ALL factorizations

• Highest Exponents: Use maximum exponent across all factorizations

• Verification: LCM must be divisible by all original numbers

LCM Methods & Techniques
LCM(a,b) = Product of all prime factors with highest exponents
LCM Formula
Method 1
Listing Multiples
Best for small numbers
Method 2
Prime Factorization
Most reliable method
Method 3
GCF Relationship
Using LCM = (a×b)/GCF(a,b)
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is divisible by two or more numbers.

Multiple: A number that can be divided by another number without remainder.

Prime Factorization: Expressing a number as a product of prime numbers.

GCF-LCM Relationship: LCM(a,b) × GCF(a,b) = a × b.

Complete methodology:
  1. Listing Multiples: For small numbers, list multiples and find the smallest common one
  2. Prime Factorization: Find prime factors of each number and multiply all primes with highest exponents
  3. GCF Relationship: Use LCM(a,b) = (a×b)/GCF(a,b) when GCF is known
  4. Verification: Check that LCM is divisible by all original numbers
Tip 1: Use prime factorization method for most problems - it's most reliable
Tip 2: For LCM of multiple numbers, include ALL primes from ALL factorizations
Tip 3: Always verify by dividing LCM by original numbers to check for whole numbers
Tip 4: Use GCF relationship: LCM(a,b) = (a×b)/GCF(a,b) when convenient
Common errors: Including only common factors (not all), using lowest instead of highest exponents, forgetting that LCM must be divisible by all numbers.
Key properties: LCM(a,b) ≥ max(a,b), LCM of coprime numbers is a×b, LCM(a,a) = a.
Essential rules:

• All Primes Required: Include every prime that appears in any factorization

• Highest Exponents: Use maximum exponent for each prime

• Verification: LCM must be divisible by each original number

• Minimum Value: LCM cannot be smaller than the largest number

Solution: Exercises 4 to 5
4 GCF-LCM Relationship
Exercise 4
Find the LCM of 24 and 36 using the GCF relationship: LCM(a,b) = (a×b)/GCF(a,b).
Definition:

GCF-LCM Relationship: LCM(a,b) × GCF(a,b) = a × b, so LCM(a,b) = (a×b)/GCF(a,b).

Numbers
24 and 36
GCF
12
LCM
(24×36)/12 = 72
Step 1: Find GCF of 24 and 36

Prime factorization of 24: 2³ × 3

Prime factorization of 36: 2² × 3²

GCF = 2² × 3¹ = 4 × 3 = 12

Step 2: Apply the formula

LCM(24, 36) = (24 × 36) ÷ GCF(24, 36)

LCM(24, 36) = (24 × 36) ÷ 12

Step 3: Calculate

24 × 36 = 864

864 ÷ 12 = 72

Step 4: Verification

72 ÷ 24 = 3 (whole number)

72 ÷ 36 = 2 (whole number)

✓ Both divisions result in whole numbers

LCM(24, 36) = 72
Final answer:

The LCM of 24 and 36 is 72.

Applied rules:

• GCF-LCM Formula: LCM(a,b) = (a×b)/GCF(a,b)

• Relationship: LCM × GCF = a × b

• Verification: LCM must be divisible by both original numbers

5 Application Problem
Exercise 5
Two buses leave a station at the same time. Bus A returns every 15 minutes, and Bus B returns every 20 minutes. After how many minutes will both buses return to the station at the same time?
Definition:

Application Problem: Real-world scenario requiring LCM to find when cycles align.

Bus Cycles
15 min and 20 min
Prime Factorizations
3×5 and 2²×5
LCM
2²×3×5 = 60 min
Step 1: Identify the problem

We need to find when both buses will be at the station simultaneously

This occurs at the LCM of their return intervals

Step 2: Find LCM of 15 and 20

Prime factorization of 15: 3 × 5

Prime factorization of 20: 2² × 5

Step 3: Identify all prime factors

Prime factors present: 2, 3, and 5

Highest exponent of 2: max(0,2) = 2

Highest exponent of 3: max(1,0) = 1

Highest exponent of 5: max(1,1) = 1

Step 4: Calculate LCM

LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Step 5: Verify

After 60 minutes:

Bus A returns: 60 ÷ 15 = 4 times (at 15, 30, 45, 60 min)

Bus B returns: 60 ÷ 20 = 3 times (at 20, 40, 60 min)

Both buses are at the station at 60 minutes

Both buses return at 60 minutes
Final answer:

Both buses will return to the station at the same time after 60 minutes.

Applied rules:

• Real-world Application: LCM finds when repeating events align

• Cycle Alignment: Events repeat at LCM of their periods

• Verification: Check that LCM is divisible by both periods

LCM Applications & Advanced Techniques
LCM(a,b) = ∏(pᵢ^(max(e₁,e₂)))
LCM Formula
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is divisible by two or more numbers.

Lowest Common Multiple: Alternative name for LCM, used in some regions.

Coprime Numbers: Numbers whose GCF is 1 (no common factors other than 1).

GCF-LCM Relationship: LCM(a,b) × GCF(a,b) = a × b.

Complete methodology:
  1. For Small Numbers: Use listing multiples method
  2. For Medium Numbers: Use prime factorization method
  3. When GCF Known: Use GCF relationship method
  4. For Multiple Numbers: Apply method iteratively
Tip 1: LCM is essential for finding common denominators in fraction addition
Tip 2: If GCF(a,b) = 1, then LCM(a,b) = a × b
Tip 3: GCF(a,b) × LCM(a,b) = a × b
Tip 4: Always verify by ensuring LCM is divisible by all original numbers
Common errors: Including only common factors (not all), using lowest instead of highest exponents, forgetting that LCM must be divisible by all numbers.
Applications: Common denominators, event synchronization, gear ratios, periodic phenomena.
Essential rules:

• All Primes Required: Include every prime that appears in any factorization

• Highest Exponents: Use maximum exponent for each prime

• Verification: LCM must be divisible by each original number

• Minimum Value: LCM cannot be smaller than the largest number

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

Questions & Answers

Question: I'm confused about when to use the different methods for finding LCM. How do I choose the best one?

Answer: Here's a guide for choosing the best LCM method:

  • Listing Multiples (Small numbers, ≤ 10): Quick and visual for small numbers
  • Prime Factorization (Medium numbers, 10-200): Most reliable method, works for any numbers
  • GCF Relationship (When GCF known): Efficient when GCF is easily calculated

For most pre-algebra problems, prime factorization is recommended because it's systematic and works well for medium-sized numbers. It also reinforces understanding of prime numbers and factorization, which are important concepts.

Question: What's the relationship between LCM and GCF? How are they connected?

Answer: There's a beautiful mathematical relationship between LCM and GCF:

LCM(a,b) × GCF(a,b) = a × b

This means:

  • LCM(a,b) = (a × b) ÷ GCF(a,b)
  • GCF(a,b) = (a × b) ÷ LCM(a,b)

For example: LCM(12,18) = 36 and GCF(12,18) = 6

Check: 36 × 6 = 216 = 12 × 18 ✓

This relationship is particularly useful when you know one and need to find the other.

Question: How is LCM related to adding and subtracting fractions with different denominators?

Answer: LCM is fundamental to adding and subtracting fractions with different denominators:

  1. Find LCM: Calculate the LCM of the denominators (this becomes the common denominator)
  2. Convert Fractions: Rewrite each fraction with the common denominator
  3. Add/Subtract: Perform the operation on the numerators

Example: Add 1/4 + 1/6

  • LCM(4, 6) = 12
  • 1/4 = 3/12 (multiply top and bottom by 3)
  • 1/6 = 2/12 (multiply top and bottom by 2)
  • 1/4 + 1/6 = 3/12 + 2/12 = 5/12

The LCM gives us the smallest possible common denominator, which keeps the numbers manageable.