Least Common Multiple (LCM) in Grade 4

Master Least Common Multiple (LCM): finding common multiples, comparing numbers, and applying LCM in word problems through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Finding the LCM of Two Numbers
Exercise 1
Find the least common multiple (LCM) of 6 and 8. Show all your work and explain your method.
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers. It's the smallest number that both numbers divide into evenly.

LCM Finding Method:
  1. List the multiples of the first number
  2. List the multiples of the second number
  3. Identify the common multiples (numbers that appear in both lists)
  4. Choose the smallest number from the common multiples
Numbers
6 and 8
Common Multiples
24, 48, 72...
LCM
24
Step 1: List 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...

Step 2: List 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 3: Identify common multiples

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64...

Common multiples: 24, 48, 72...

Step 4: Select the least common multiple

From 24, 48, 72... → the smallest is 24

LCM(6, 8) = 24
Final answer:

The least common multiple of 6 and 8 is 24.

We know this is correct because 24÷6=4 and 24÷8=3 (both divide evenly).

Applied rules:

• Multiple Definition: A multiple of a number is the product of that number and an integer

• Common Multiple Rule: A common multiple is divisible by both numbers

• Least Rule: The LCM is the smallest number that both original numbers divide into

Tip 1: Continue listing multiples until you find a match in both lists.
Tip 2: The LCM is always greater than or equal to the larger of the two numbers.
2 Applying LCM to Real-World Problems
Exercise 2
Two buses leave the station at the same time. One bus returns every 12 minutes, and the other returns every 18 minutes. After how many minutes will both buses return to the station at the same time again?
Definition:

Real-World Application: LCM helps determine when two recurring events will align or occur simultaneously.

Bus 1 Interval
12 minutes
Bus 2 Interval
18 minutes
LCM
36 minutes
Step 1: Identify the intervals to compare

Bus 1 returns every 12 minutes

Bus 2 returns every 18 minutes

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

Step 2: List multiples of 12

12×1=12, 12×2=24, 12×3=36, 12×4=48, 12×5=60

Multiples of 12: 12, 24, 36, 48, 60...

Step 3: List multiples of 18

18×1=18, 18×2=36, 18×3=54, 18×4=72, 18×5=90

Multiples of 18: 18, 36, 54, 72, 90...

Step 4: Identify the first common multiple

Multiples of 12: 12, 24, 36, 48, 60...

Multiples of 18: 18, 36, 54, 72, 90...

First common multiple: 36

Step 5: Verify the answer

36÷12=3 (bus 1 returns 3 times)

36÷18=2 (bus 2 returns 2 times)

Both buses return at 36 minutes

Both buses return at 36 minutes
Final answer:

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

Verification: Bus 1 returns 3 times (12×3=36), Bus 2 returns 2 times (18×2=36).

Applied rules:

• Recurring Events: LCM finds when cycles align

• Time Calculation: Smallest interval where both cycles complete

• Verification: Check that both numbers divide the LCM evenly

Tip 1: In scheduling problems, LCM tells you when events coincide.
Tip 2: Always verify by checking that the LCM is divisible by both original numbers.
3 Finding LCM of Three Numbers
Exercise 3
Find the least common multiple of 4, 6, and 8. Explain your process step by step.
Definition:

LCM of Multiple Numbers: The smallest number that is a multiple of ALL given numbers. The process extends the two-number method to include all numbers.

Numbers
4, 6, 8
Common Multiples
24, 48, 72...
LCM
24
Step 1: List multiples of 4

4×1=4, 4×2=8, 4×3=12, 4×4=16, 4×5=20, 4×6=24, 4×7=28, 4×8=32

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32...

Step 2: List 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...

Step 3: List 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 4: Identify multiples common to ALL three numbers

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...

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

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

Common to all: 24, 48, 72...

Step 5: Select the least common multiple

From 24, 48, 72... → the smallest is 24

LCM(4, 6, 8) = 24
Final answer:

The least common multiple of 4, 6, and 8 is 24.

Verification: 24÷4=6, 24÷6=4, 24÷8=3 (all divide evenly)

Applied rules:

• Extension Rule: LCM process works for any number of values

• Universal Multiple: LCM must be divisible by ALL given numbers

• Hierarchical Process: Find multiples of each number, then find intersection

Tip 1: When finding LCM of multiple numbers, look for the intersection of all multiple sets.
Tip 2: The LCM of multiple numbers is always ≥ the largest number.
Detailed Summary: Least Common Multiple (LCM)
\(\text{LCM}(a, b) = \text{smallest number divisible by both } a \text{ and } b\)
LCM Formula
Key Concepts and Definitions:

Multiple: The product of a number and an integer (e.g., multiples of 3 are 3, 6, 9, 12...)

Common Multiple: A number that is a multiple of two or more numbers (e.g., 12 is a common multiple of 3 and 4)

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers (e.g., LCM of 3 and 4 is 12)

Prime Factor: A factor that is also a prime number (used in alternative LCM methods)

Complete LCM Finding Method:
  1. List multiples: Find multiples of each number by multiplying systematically
  2. Compare lists: Identify which multiples appear in all lists
  3. Select least: From the common multiples, choose the smallest one
  4. Verify: Check that the LCM is divisible by all original numbers
Tip 1: Start listing multiples with the larger number to find matches faster.
Tip 2: The LCM is always ≥ the largest of the compared numbers.
Tip 3: If one number divides the other evenly, the LCM is the larger number.
Tip 4: For consecutive numbers, LCM is often their product.
Tip 5: In word problems, LCM often represents when events align.
Tip 6: For large numbers, consider using prime factorization method.
Common errors: Stopping too early before finding a common multiple, listing multiples incorrectly, selecting a common multiple that isn't the least.
Memorization tips: LCM = Least Common Multiple = Smallest number that all others divide into evenly. Think "What's the smallest number both can fit into?"
Core Rules and Properties:

• Reflexive Property: LCM(a, a) = a

• Commutative Property: LCM(a, b) = LCM(b, a)

• Bounded Property: LCM(a, b) ≥ max(a, b)

• Divisibility Property: If m = LCM(a, b), then both a and b divide m

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

LCM Examples
🔢
Simple Example

LCM(4, 6):

Multiples of 4: 4, 8, 12, 16, 20...

Multiples of 6: 6, 12, 18, 24...

Common: 12 → LCM = 12

Complex Example

LCM(12, 18, 24):

Requires finding multiples of all three numbers

Systematic approach necessary

LCM = 72

Special Case

LCM of coprime numbers:

Numbers with GCF = 1

LCM = Product of numbers

LCM Problem Solving Steps
1
List Multiples
2
Find Common
3
Select LCM
4
Verify
Venn Diagram Representation
Multiples of A ∩ Multiples of B = Common Multiples
Remember: Systematic approach wins!
Advanced LCM Techniques and Applications
\(\text{LCM}(a, b) = \min\{m : a|m \land b|m\}\)
Mathematical Notation
Method 1
Listing Multiples
Most Common Approach
Method 2
Prime Factorization
For Larger Numbers
Method 3
Using GCF
Advanced Technique
Advanced Concepts:

Coprime Numbers: Two numbers with GCF = 1 (LCM = their product)

Relatively Prime: Another term for coprime numbers

Prime Factorization: Breaking numbers into prime factors to find LCM

GCF Connection: LCM(a,b) = (a×b)/GCF(a,b)

Alternative Methods:

Prime Factorization Method: Break each number into prime factors, then multiply the highest power of all primes.

Example: LCM(12, 18)

12 = 2² × 3, 18 = 2 × 3²

Take highest powers: 2² and 3² → LCM = 2²×3² = 4×9 = 36

Quick Check: If one number divides the other evenly, the LCM is the larger number.
Consecutive Numbers: LCM of n and (n+1) is n(n+1).
Coprime Numbers: If GCF(a,b)=1, then LCM(a,b)=a×b.
Word Problem Clue: "When will they meet again?" or "next time together" usually indicates LCM.
Applications: Scheduling, gear ratios, finding common denominators, periodic events.
Connections: LCM connects to GCF, fraction addition, and number theory fundamentals.
Important Relationships:

• Fraction Addition: LCM of denominators gives the common denominator

• Periodic Events: LCM determines when recurring events align

• Algebra Connection: Finding common denominators uses LCM concept

• Number Theory: LCM is fundamental to understanding divisibility

Questions & Answers

Question: How do I know when to stop listing multiples? It seems like I might have to go on forever!

Answer: You'll always find a common multiple eventually! Here's how to approach it efficiently:

  • Start with the larger number: List multiples of the larger number first, as you're more likely to find a match sooner
  • Look for matches: As you list multiples of the larger number, check if they appear in the smaller number's multiples
  • Stop at the first match: Once you find a number that appears in both lists, that's your LCM!
  • For 3+ numbers: Find LCM of first two, then find LCM of that result with the third number

Example: LCM(6, 9)

Multiples of 9: 9, 18, 27, 36...

Check if any are multiples of 6: 9? No. 18? Yes! 18÷6=3

So LCM(6, 9) = 18

You don't need to continue listing multiples once you find the first common one!

Question: Is there a connection between LCM and GCF? They seem like opposite concepts.

Answer: Yes, there's a fascinating relationship between LCM and GCF! They're actually connected by this beautiful formula:

LCM(a, b) × GCF(a, b) = a × b

This means you can find one if you know the other:

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

Example: For numbers 12 and 18

GCF(12, 18) = 6 (we found this earlier)

So LCM(12, 18) = (12 × 18) ÷ 6 = 216 ÷ 6 = 36

Let's verify: LCM(12, 18) = 36 (since 36÷12=3 and 36÷18=2)

This confirms our relationship: 36 × 6 = 12 × 18 = 216 ✓

They complement each other beautifully in number theory!

Question: Why is finding the LCM useful in real life? When would I ever need to use it?

Answer: LCM has many practical applications! Here are some real-world scenarios:

  • Scheduling: Finding when two repeating events will align (like our bus example)
  • Cooking: Scaling recipes to serve exact number of people using whole ingredients
  • Construction: Finding smallest length where two different measurement units align
  • Music: Finding when two rhythms with different beats per measure will sync
  • Mathematics: Adding fractions with different denominators

Example: If you need to buy packs of 6 cookies and 8 cookies, and want the same number of each, LCM(6,8)=24 tells you to buy 4 packs of 6 (24 cookies) and 3 packs of 8 (24 cookies).

LCM helps find the smallest common solution when dealing with different repeating cycles!