Worksheet on Finding the Least Common Multiple of Two Numbers in Grade 4

Master finding the least common multiple of two numbers: listing method, prime factorization, and GCD method through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Listing Method
Exercise 1
Find the LCM of 4 and 6 using the listing method.
Definition:

Least Common Multiple (LCM): The smallest number that is a multiple of both numbers.

Listing Method:
  1. List the multiples of the first number
  2. List the multiples of the second number
  3. Find the smallest number that appears in both lists
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Step 1: List multiples of 4

4, 8, 12, 16, 20, 24, 28, 32, 36...

Step 2: List multiples of 6

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

Step 3: Find common multiples

12 appears in both lists

24 appears in both lists

36 appears in both lists

Step 4: Identify the smallest

The smallest common multiple is 12

12
24
36
LCM of 4 and 6 = 12
LCM(4, 6) = 12
Final answer:

The LCM of 4 and 6 is 12

Applied rules:

• Common multiple: A number that appears in both lists

• Least: The smallest of all common multiples

• Verification: 12 ÷ 4 = 3 and 12 ÷ 6 = 2 (both whole numbers)

2 Another Listing Example
Exercise 2
Find the LCM of 3 and 5 using the listing method.
Definition:

Least Common Multiple (LCM): The smallest number that is a multiple of both numbers.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35...
Step 1: List multiples of 3

3, 6, 9, 12, 15, 18, 21, 24, 27, 30...

Step 2: List multiples of 5

5, 10, 15, 20, 25, 30, 35...

Step 3: Find common multiples

15 appears in both lists

30 appears in both lists

Step 4: Identify the smallest

The smallest common multiple is 15

15
30
LCM of 3 and 5 = 15
LCM(3, 5) = 15
Final answer:

The LCM of 3 and 5 is 15

Applied rules:

• Prime numbers: 3 and 5 are both prime, so their LCM is their product

• Common multiple: A number that appears in both lists

• Verification: 15 ÷ 3 = 5 and 15 ÷ 5 = 3 (both whole numbers)

3 Larger Numbers
Exercise 3
Find the LCM of 6 and 8 using the listing method.
Definition:

Least Common Multiple (LCM): The smallest number that is a multiple of both numbers.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56...
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: Find common multiples

24 appears in both lists

48 appears in both lists

Step 4: Identify the smallest

The smallest common multiple is 24

24
48
LCM of 6 and 8 = 24
LCM(6, 8) = 24
Final answer:

The LCM of 6 and 8 is 24

Applied rules:

• Common multiple: A number that appears in both lists

• Least: The smallest of all common multiples

• Verification: 24 ÷ 6 = 4 and 24 ÷ 8 = 3 (both whole numbers)

LCM Rules and Methods
LCM(a,b) = \frac{a × b}{GCD(a,b)}
LCM Formula
Rule 1
LCM(a,b) = smallest\ common\ multiple
Definition of LCM
Rule 2
LCM(a,a) = a
LCM of same number
Rule 3
LCM(a,1) = a
LCM with 1
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is divisible by both numbers

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

Common Multiple: A number that is a multiple of two or more numbers

Greatest Common Divisor (GCD): The largest number that divides both numbers evenly

LCM Finding Methods:
  1. Listing Method: List multiples of each number, find the smallest common one
  2. Prime Factorization: Find prime factors of each number, take highest power of each prime
  3. GCD Formula: Use the formula LCM(a,b) = (a×b) ÷ GCD(a,b)
  4. Verification: Check that the LCM is divisible by both original numbers
Tip 1: For small numbers, the listing method is easiest.
Tip 2: If one number is a multiple of the other, the larger is the LCM.
Tip 3: If two numbers are coprime (GCD = 1), their LCM is their product.
Tip 4: Always verify by dividing the LCM by both original numbers.
LCM patterns: LCM is always ≥ the larger of the two numbers.
Key insight: LCM helps find common denominators and solve scheduling problems.
LCM Properties:

• LCM(a, b) ≥ max(a, b)

• LCM(a, b) = LCM(b, a) (commutative)

• LCM(a, a) = a

• LCM(a, 1) = a

Solution: Exercises 4 to 5
4 Special Case
Exercise 4
Find the LCM of 4 and 12 using the listing method.
Definition:

Special Case: When one number is a multiple of the other, the LCM is the larger number.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
Multiples of 12: 12, 24, 36, 48, 60...
Step 1: List multiples of 4

4, 8, 12, 16, 20, 24, 28, 32, 36...

Step 2: List multiples of 12

12, 24, 36, 48, 60...

Step 3: Find common multiples

12 appears in both lists

24 appears in both lists

36 appears in both lists

Step 4: Identify the smallest

The smallest common multiple is 12

Step 5: Recognize the pattern

Since 12 is a multiple of 4 (12 = 4 × 3), the LCM is 12

12
24
36
LCM of 4 and 12 = 12
LCM(4, 12) = 12
Final answer:

The LCM of 4 and 12 is 12

Applied rules:

• Special case: When one number divides the other evenly, the larger is the LCM

• Verification: 12 ÷ 4 = 3 and 12 ÷ 12 = 1 (both whole numbers)

• Pattern recognition: Look for divisibility relationships

5 Three Numbers Challenge
Exercise 5
Find the LCM of 2, 3, and 4 using the listing method.
Definition:

LCM of Three Numbers: The smallest number that is divisible by all three numbers.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
Step 1: List multiples of 2

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...

Step 2: List multiples of 3

3, 6, 9, 12, 15, 18, 21, 24, 27, 30...

Step 3: List multiples of 4

4, 8, 12, 16, 20, 24, 28, 32, 36...

Step 4: Find common multiples

Looking for numbers that appear in all three lists:

12 appears in all lists

24 appears in all lists

Step 5: Identify the smallest

The smallest number that appears in all three lists is 12

12
24
LCM of 2, 3, and 4 = 12
LCM(2, 3, 4) = 12
Final answer:

The LCM of 2, 3, and 4 is 12

Applied rules:

• Extension: LCM method works for any number of numbers

• Common multiple: Must appear in all lists

• Verification: 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3 (all whole numbers)

LCM Complete Guide
LCM(a,b) = \frac{a × b}{GCD(a,b)}
LCM Formula
Key definitions:

Least Common Multiple (LCM): The smallest positive integer that is divisible by both numbers

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

Common Multiple: A number that is a multiple of two or more numbers

Greatest Common Divisor (GCD): The largest number that divides both numbers evenly

LCM Finding Methods:
  1. Listing Method: List multiples of each number, find the smallest common one
  2. Prime Factorization: Find prime factors of each number, take highest power of each prime
  3. GCD Formula: Use the formula LCM(a,b) = (a×b) ÷ GCD(a,b)
  4. Verification: Check that the LCM is divisible by both original numbers
Tip 1: For small numbers, the listing method is easiest.
Tip 2: If one number is a multiple of the other, the larger is the LCM.
Tip 3: If two numbers are coprime (GCD = 1), their LCM is their product.
Tip 4: Always verify by dividing the LCM by both original numbers.
Tip 5: LCM is useful for finding common denominators in fractions.
LCM patterns: LCM is always ≥ the larger of the two numbers, LCM(a,b) = LCM(b,a), LCM(a,a) = a.
Key insight: LCM helps find common denominators and solve scheduling problems.
Important Properties:

• LCM(a, b) ≥ max(a, b)

• LCM(a, b) = LCM(b, a) (commutative)

• LCM(a, a) = a

• LCM(a, 1) = a

• LCM(a, b) × GCD(a, b) = a × b

LCM Finding Patterns
📊
Listing Method
List multiples, find smallest common
4: 4,8,12,16,20,24
6: 6,12,18,24
LCM = 12
Special Cases
One divides other → larger is LCM
4 divides 12
LCM(4,12)=12
Coprime Numbers
No common factors → LCM = product
3 and 5 coprime
LCM(3,5)=15
LCM Finding Steps
1
List multiples
2
Find common
3
Identify smallest
4
Verify
LCM Examples
LCM(4,6) = 12
LCM(3,5) = 15
LCM(6,8) = 24
LCM(4,12) = 12
Remember: LCM is the Smallest Common Multiple!

Questions & Answers

Question: How do I know if I've found the least common multiple or just a common multiple?

Answer: The "least" part means the smallest number that appears in both lists. Here's how to be sure:

  • Compare all common multiples: If you find 12, 24, 36 as common multiples, 12 is the least
  • Check your lists: Make sure you listed enough multiples to find the first one that appears in both
  • Verify divisibility: The LCM should divide evenly by both original numbers

For example, with 4 and 6: multiples of 4 are 4, 8, 12, 16, 20, 24... and multiples of 6 are 6, 12, 18, 24... The first number that appears in both is 12, so that's the LCM.

If you find 24 first, keep looking earlier in your lists - there might be a smaller common multiple!

Question: Is there a faster way to find the LCM without listing all the multiples?

Answer: Yes! For now, the listing method works well for small numbers. Later you'll learn other methods:

  • Special cases: If one number divides the other, the larger is the LCM
  • Prime factorization: Find prime factors and take the highest power of each
  • Pattern recognition: For coprime numbers (no common factors), LCM = product of the numbers

For example: LCM(4, 12) = 12 (since 12 divides 4 evenly), LCM(3, 5) = 15 (since 3 and 5 are coprime).

The listing method is the most reliable for now, but recognizing these patterns will make you faster!

Question: What if I need to find the LCM of three numbers? Is it the same method?

Answer: Yes! The method extends to three or more numbers:

  1. List multiples of each number just like with two numbers
  2. Find numbers that appear in ALL lists (not just some of them)
  3. Choose the smallest number that appears in all lists
  4. Verify: Check that this number divides evenly by all original numbers

For example, with 2, 3, and 4: multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...; multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24...; multiples of 4 are 4, 8, 12, 16, 20, 24... The first number in all three lists is 12, so LCM(2, 3, 4) = 12.

It takes more work but follows the same principle!