Solved Exercises on Understanding Factors and Multiples in Grade 4

Master factors and multiples: finding factors, multiples, prime numbers, common factors, and least common multiples through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding factors
Exercise 1
Find all the factors of 24. List them in order from least to greatest.
Definition:

Factor: A number that divides evenly into another number

Factor pair: Two numbers that multiply to give the original number

Factor-finding method:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. For each divisor found, record both the divisor and its pair
  4. List all factors in order
Number to factor
24
Factor pairs
1×24, 2×12, 3×8, 4×6
All factors
1, 2, 3, 4, 6, 8, 12, 24
Step 1: Start with 1 and 24

1 × 24 = 24, so 1 and 24 are factors

Step 2: Check 2

24 ÷ 2 = 12, so 2 and 12 are factors

Step 3: Check 3

24 ÷ 3 = 8, so 3 and 8 are factors

Step 4: Check 4

24 ÷ 4 = 6, so 4 and 6 are factors

Step 5: Check 5

24 ÷ 5 = 4.8 (not a whole number), so 5 is not a factor

Step 6: Check 6

We already have 6 as a factor, so stop here

1
2
3
4
6
8
12
24
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

Applied rules:

• Factor definition: A factor divides the number evenly with no remainder

• Factor pairs: Factors come in pairs that multiply to the original number

• Systematic approach: Check numbers in order to ensure no factors are missed

2 Finding multiples
Exercise 2
List the first 6 multiples of 7.
Definition:

Multiple: The product of a number and any whole number

Examples: Multiples of 7 are 7×1, 7×2, 7×3, etc.

Base number
7
First 6 multiples
7, 14, 21, 28, 35, 42
Step 1: Multiply by 1

7 × 1 = 7

Step 2: Multiply by 2

7 × 2 = 14

Step 3: Multiply by 3

7 × 3 = 21

Step 4: Multiply by 4

7 × 4 = 28

Step 5: Multiply by 5

7 × 5 = 35

Step 6: Multiply by 6

7 × 6 = 42

7
14
21
28
35
42
First 6 multiples of 7: 7, 14, 21, 28, 35, 42
Final answer:

The first 6 multiples of 7 are: 7, 14, 21, 28, 35, 42

Applied rules:

• Multiple definition: A multiple is the result of multiplying a number by whole numbers

• Pattern: Multiples form an arithmetic sequence with the base number as the common difference

• Infinite set: There are infinitely many multiples of any number

3 Prime and composite numbers
Exercise 3
Is 17 a prime number or a composite number? Explain your answer.
Definition:

Prime number: A number greater than 1 that has exactly two factors: 1 and itself

Composite number: A number greater than 1 that has more than two factors

Number to check
17
Factors of 17
1, 17
Classification
Prime
Step 1: Check if 17 is greater than 1

Yes, 17 > 1

Step 2: Find all factors of 17

Check divisibility by numbers from 2 to √17 (≈4.1)

17 ÷ 2 = 8.5 (not a factor)

17 ÷ 3 = 5.67 (not a factor)

17 ÷ 4 = 4.25 (not a factor)

Only 1 and 17 divide 17 evenly

Step 3: Count the factors

17 has exactly 2 factors: 1 and 17

Step 4: Classify the number

Since 17 has exactly two factors, it is a prime number

1
17
17 is a prime number
Final answer:

17 is a prime number because it has exactly two factors: 1 and 17.

Applied rules:

• Prime number: Has exactly two distinct positive divisors

• Composite number: Has more than two distinct positive divisors

• Checking method: Only check divisors up to the square root of the number

Factors and Multiples Review
\(\text{Factor} \times \text{Factor} = \text{Product}\)
Factor Relationship
Key definitions:

Factor: A number that divides another number evenly with no remainder

Multiple: The result of multiplying a number by a whole number

Prime number: A number with exactly two factors: 1 and itself

Composite number: A number with more than two factors

Greatest Common Factor (GCF): The largest factor shared by two or more numbers

Least Common Multiple (LCM): The smallest multiple shared by two or more numbers

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

Factor-finding strategies:
  1. Start systematically: Begin with 1 and the number itself
  2. Check sequentially: Test each number up to the square root
  3. Record pairs: When you find a factor, record both numbers in the pair
  4. Stop efficiently: Stop when you reach a repeated factor
Tip 1: Every number has at least two factors: 1 and itself.
Tip 2: To find multiples, multiply the number by 1, 2, 3, 4...
Tip 3: The largest factor of a number is the number itself.
Tip 4: The smallest multiple of a number is the number itself.
Common errors: Confusing factors and multiples, missing factor pairs, misidentifying primes.
Divisibility tests: Use shortcuts to identify factors quickly.
Divisibility rules:

• Divisible by 2: Number ends in 0, 2, 4, 6, or 8

• Divisible by 3: Sum of digits is divisible by 3

• Divisible by 5: Number ends in 0 or 5

• Divisible by 9: Sum of digits is divisible by 9

• Divisible by 10: Number ends in 0

Prime vs Composite Numbers

Prime Numbers

Have exactly 2 factors: 1 and itself

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23

2 is the only even prime number

Composite Numbers

Have more than 2 factors

Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20

Can be factored into smaller numbers

Solution: Exercises 4 to 5
4 Common factors
Exercise 4
Find the common factors of 18 and 24. What is the greatest common factor (GCF)?
Definition:

Common factors: Factors that two or more numbers share

Greatest Common Factor (GCF): The largest of the common factors

Factors of 18
1, 2, 3, 6, 9, 18
Factors of 24
1, 2, 3, 4, 6, 8, 12, 24
Common factors
1, 2, 3, 6
Step 1: Find factors of 18

18: 1×18, 2×9, 3×6

Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: Find factors of 24

24: 1×24, 2×12, 3×8, 4×6

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 3: Identify common factors

Look for factors that appear in both lists

Common factors: 1, 2, 3, 6

Step 4: Find the greatest common factor

From common factors 1, 2, 3, 6, the largest is 6

GCF = 6

1
2
3
6
Common factors: 1, 2, 3, 6
GCF: 6
Final answer:

The common factors of 18 and 24 are: 1, 2, 3, 6

The Greatest Common Factor (GCF) is 6.

Applied rules:

• Common factors: Identify factors that appear in both factor lists

• GCF: The largest number that divides both original numbers

• Systematic approach: Organize factor lists to easily identify overlaps

5 Least common multiple
Exercise 5
Find the least common multiple (LCM) of 6 and 8.
Definition:

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

Multiples of 6
6, 12, 18, 24, 30, 36...
Multiples of 8
8, 16, 24, 32, 40...
LCM
24
Step 1: List multiples of 6

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

Step 2: List multiples of 8

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

Step 3: Find the first common multiple

Looking at both lists: 6, 12, 18, 24, 30, 36... and 8, 16, 24, 32, 40...

The first number that appears in both lists is 24

Step 4: State the LCM

The least common multiple of 6 and 8 is 24

6
12
18
24
and
8
16
24
LCM of 6 and 8: 24
Final answer:

The least common multiple of 6 and 8 is 24.

Applied rules:

• LCM definition: Smallest number that both original numbers divide into evenly

• Method: List multiples of each number until a common one is found

• Alternative method: Use prime factorization for larger numbers

Comprehensive Factors and Multiples Guide
\(\text{LCM}(a,b) = \frac{a \times b}{\text{GCF}(a,b)}\)
LCM Formula
Key definitions:

Factor: A number that divides another number evenly with no remainder

Multiple: The result of multiplying a number by a whole number

Prime number: A number with exactly two factors: 1 and itself

Composite number: A number with more than two factors

Greatest Common Factor (GCF): The largest factor shared by two or more numbers

Least Common Multiple (LCM): The smallest multiple shared by two or more numbers

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

Finding GCF and LCM methods:
  1. List method: List all factors/multiples and identify common ones
  2. Prime factorization: Break numbers into prime factors
  3. Division method: Use the Euclidean algorithm for GCF
Tip 1: The GCF is always less than or equal to the smaller number.
Tip 2: The LCM is always greater than or equal to the larger number.
Tip 3: Use divisibility rules to quickly identify factors.
Tip 4: For large numbers, use prime factorization to find GCF and LCM.
Special cases: 1 is neither prime nor composite; 2 is the only even prime.
Relationship: For any two numbers, GCF × LCM = product of the numbers.
Important relationships:

• Factor-multiple relationship: If a is a factor of b, then b is a multiple of a

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

• Prime factorization: Every composite number can be written uniquely as a product of primes

• Even numbers: All even numbers (except 2) are composite

Prime Factorization Tree

Example: Factor 24
24
2 × 12
2 × 2 × 6
2 × 2 × 2 × 3

Prime factorization: 24 = 2³ × 3

Prime Numbers (2-30)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

2 3 5 7 11 13 17 19 23 29

Questions & Answers

Question: I'm confused about the difference between factors and multiples. Can you explain it more clearly?

Answer: Great question! Think of it this way:

  • Factors: These are the numbers that divide INTO another number. They are usually smaller than or equal to the number. For example, factors of 12 are 1, 2, 3, 4, 6, 12.
  • Multiples: These are the numbers you get WHEN you multiply a number. They are usually larger than or equal to the number. For example, multiples of 3 are 3, 6, 9, 12, 15, 18...

A helpful memory trick: FFactors go into the number (they divide it), and MMultiples come from the number (they are made by multiplying it).

Also remember: there are finite factors but infinite multiples!

Question: My child is having trouble identifying prime numbers. What's the easiest way to teach this concept?

Answer: Here's a simple approach to identify prime numbers:

  • Definition: A prime number has exactly two factors: 1 and itself
  • Quick check: Test divisibility by small primes (2, 3, 5, 7) up to the square root of the number
  • Memory aids: The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Teach your child to eliminate obvious composites first:

  • All even numbers (except 2) are composite
  • Numbers ending in 5 (except 5) are composite
  • Use divisibility rules for 3 (add digits, if sum is divisible by 3, so is the number)

Remember: 1 is neither prime nor composite!

Question: How can I help students remember the difference between GCF and LCM?

Answer: Here are effective memory aids:

  • GCF (Greatest Common Factor): "Think of the Greatest factor that Commonly divides both numbers"
  • LCM (Least Common Multiple): "Think of the Least multiple that both numbers Commonly multiply into"

Visual approach:

  • GCF: Focus on what divides the numbers (factors are usually smaller)
  • LCM: Focus on what the numbers divide into (multiples are usually larger)

Also remember: GCF ≤ smaller number, and LCM ≥ larger number!

Practice with real-world problems like "When will two events happen together again?" (LCM) versus "What's the largest group size that works for both?" (GCF).