Solved Exercises on Multiples and Factors in Grade 4

Master multiples and factors: definitions, rules, methods, and detailed step-by-step solutions through these 5 comprehensive exercises.

Solution: Exercises 1 to 3
1 Finding factors
Exercise 1
List all factors of 24.
Definition:

Factor: A number that divides another number evenly without remainder

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

Factor finding method:
  1. Start with 1 and the number itself
  2. Test each number from 2 up to the square root of the number
  3. If a number divides evenly, record both the divisor and quotient
  4. List all factors in ascending order
Number
24
Factor pairs
1×24, 2×12, 3×8, 4×6
Step 1: Start with 1 and 24

1 × 24 = 24, so 1 and 24 are factors

Step 2: Test 2

24 ÷ 2 = 12, so 2 and 12 are factors

Step 3: Test 3

24 ÷ 3 = 8, so 3 and 8 are factors

Step 4: Test 4

24 ÷ 4 = 6, so 4 and 6 are factors

Step 5: Test 5

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

Step 6: Stop at √24 ≈ 4.9

We've found all factor pairs

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:

• Division: A factor divides the number evenly

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

• Efficiency: Only test up to the square root of the number

2 Identifying multiples
Exercise 2
Which numbers are multiples of 7 from this list: 14, 21, 28, 35, 42, 48?
Definition:

Multiple: A number that results from multiplying a given number by a whole number

Examples: Multiples of 7 are: 7×1=7, 7×2=14, 7×3=21, etc.

Number
7
List
14, 21, 28, 35, 42, 48
Multiples
14, 21, 28, 35, 42
Step 1: Check 14

14 ÷ 7 = 2 (whole number), so 14 is a multiple of 7

Step 2: Check 21

21 ÷ 7 = 3 (whole number), so 21 is a multiple of 7

Step 3: Check 28

28 ÷ 7 = 4 (whole number), so 28 is a multiple of 7

Step 4: Check 35

35 ÷ 7 = 5 (whole number), so 35 is a multiple of 7

Step 5: Check 42

42 ÷ 7 = 6 (whole number), so 42 is a multiple of 7

Step 6: Check 48

48 ÷ 7 = 6.857... (not a whole number), so 48 is NOT a multiple of 7

Multiples of 7: 14, 21, 28, 35, 42
Final answer:

The multiples of 7 from the list are: 14, 21, 28, 35, 42

Applied rules:

• Division test: If division results in a whole number, it's a multiple

• Multiplication pattern: Multiples follow the pattern: n×1, n×2, n×3...

• Remainder check: If there's a remainder, it's not a multiple

3 Prime vs composite
Exercise 3
Determine if 17 is prime or composite. Explain your reasoning.
Definition:

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

Composite number: A number greater than 1 with more than two factors

Number
17
Type
Prime
Factors
1, 17
Step 1: Test divisibility by 2

17 ÷ 2 = 8.5 (not a whole number), so 2 is not a factor

Step 2: Test divisibility by 3

17 ÷ 3 = 5.67... (not a whole number), so 3 is not a factor

Step 3: Test divisibility by 4

17 ÷ 4 = 4.25 (not a whole number), so 4 is not a factor

Step 4: Test divisibility by 5

17 ÷ 5 = 3.4 (not a whole number), so 5 is not a factor

Step 5: Stop at √17 ≈ 4.1

No factors found between 2 and 4, so only factors are 1 and 17

Step 6: Classify the number

Since 17 has exactly two factors (1 and 17), it is prime

17 is a prime number because its only factors are 1 and 17
Final answer:

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

Applied rules:

• Prime definition: Exactly two factors (1 and itself)

• Composite definition: More than two factors

• Efficiency: Only test up to the square root of the number

Rules and methods, laws,...
Factor × Factor = Product
Factor Relationship
Factor Definition
If a ÷ b = c (no remainder)
Then b is a factor of a
Multiple Definition
If a = b × c
Then a is a multiple of b
Prime Number
Exactly 2 factors: 1 and itself
Example: 2, 3, 5, 7, 11, 13, 17...
Key definitions:

Factors: Numbers that divide evenly into another number

Multiples: Products of a number and any whole number

Prime: Number with exactly two factors (1 and itself)

Composite: Number with more than two factors

Factor finding method:
  1. Start with 1 and the number itself
  2. Test each number from 2 up to the square root
  3. If it divides evenly, record both numbers
  4. List all factors in ascending order
Tip 1: To find multiples, just count by the number (skip counting).
Tip 2: Every number is both a factor and multiple of itself.
Tip 3: 1 is a factor of every number.
Tip 4: The smallest factor is always 1, largest is the number itself.
Common errors: Confusing factors with multiples, forgetting 1 and the number itself.
Memory trick: Factors go INTO the number, multiples come OUT of the number.
Important rules to remember:

• Every number has at least two factors: 1 and itself

• The first few multiples of any number are found by skip counting

• Prime numbers have exactly two factors

• Composite numbers have more than two factors

• The number 1 is neither prime nor composite

Solution: Exercises 4 to 5
4 Common factors
Exercise 4
Find the common factors of 12 and 18.
Definition:

Common factors: Factors that two or more numbers share

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

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

1×12, 2×6, 3×4 → Factors: 1, 2, 3, 4, 6, 12

Step 2: Find factors of 18

1×18, 2×9, 3×6 → Factors: 1, 2, 3, 6, 9, 18

Step 3: Compare the lists

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

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

Common factors: 1, 2, 3, 6

Step 4: Identify the GCF

The greatest common factor is 6

Common factors of 12 and 18: 1, 2, 3, 6
Final answer:

The common factors of 12 and 18 are: 1, 2, 3, 6

Applied rules:

• Systematic approach: Find factors of each number separately

• Comparison: Look for factors that appear in both lists

• Organization: List factors in ascending order to avoid missing any

5 Least Common Multiple
Exercise 5
Find the LCM of 6 and 8.
Definition:

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

Numbers
6 and 8
Multiples of 6
6, 12, 18, 24, 30...
Multiples of 8
8, 16, 24, 32, 40...
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

Step 2: List multiples of 8

8×1=8, 8×2=16, 8×3=24, 8×4=32, 8×5=40, 8×6=48

Step 3: Find the first common multiple

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

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

The first common multiple is 24

Step 4: Verify the answer

24 ÷ 6 = 4 and 24 ÷ 8 = 3, so 24 is divisible by both 6 and 8

LCM of 6 and 8 is 24
Final answer:

The LCM of 6 and 8 is 24

Applied rules:

• List method: Write out multiples until you find a match

• Verification: Check that the LCM is divisible by both numbers

• Efficiency: Stop as soon as you find the first common multiple

Multiples and Factors Laws, Methods, Rules, Definitions
Factor × Factor = Product
Factor-Multiple Relationship
Key definitions:

Factor: A number that divides another number evenly without remainder

Multiple: A number that results from multiplying a given number by a whole number

Prime Number: A number greater than 1 with exactly two factors: 1 and itself

Composite Number: A number greater than 1 with more than two factors

Common Factor: A factor shared by two or more numbers

Common Multiple: A multiple shared by two or more numbers

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

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

Complete methodology:
  1. Factor finding: Test divisibility from 1 to the square root of the number
  2. Multiple identification: Check if division results in a whole number
  3. Prime testing: Check divisibility up to the square root of the number
  4. Common factors: Find factors of each number and identify overlaps
  5. LCM finding: List multiples until finding the first common one
  6. Verification: Always check your answers by multiplying or dividing
Tip 1: Factors are always smaller than or equal to the number (except for 1).
Tip 2: Multiples are always greater than or equal to the number.
Tip 3: To find multiples quickly, count by the number (skip counting).
Tip 4: The GCF is always less than or equal to the smaller number.
Tip 5: The LCM is always greater than or equal to the larger number.
Tip 6: For LCM of two numbers, if they share no common factors, multiply them.
Common errors: Confusing factors with multiples, forgetting to check divisibility by 1 and the number itself.
Key insight: Every number is a factor of itself and a multiple of itself.
Memory aid: Factors go INTO the number, multiples come OUT of the number.
Pattern: Prime numbers have exactly two factors, composites have more than two.
Essential rules to know:

• Every number has at least two factors: 1 and itself

• The number 1 is a factor of every number

• The number 0 has infinitely many factors (every number divides 0)

• Prime numbers have exactly two factors: 1 and themselves

• Composite numbers have more than two factors

• The number 1 is neither prime nor composite

• Even numbers end in 0, 2, 4, 6, or 8

• Odd numbers end in 1, 3, 5, 7, or 9

• Divisibility by 2: number ends in even digit

• Divisibility by 3: sum of digits is divisible by 3

• Divisibility by 5: number ends in 0 or 5

• Divisibility by 10: number ends in 0

Visual Learning: Factor Trees and Patterns
Exercise 6: Factor Trees
Create a factor tree for 36 and identify all factors.

Factor Tree Process: Break down a number into its prime factors by continuously dividing by prime numbers.

  • Start with 36
  • 36 = 4 × 9
  • 4 = 2 × 2, 9 = 3 × 3
  • Prime factorization: 2² × 3²
  • All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Questions & Answers

Question: I don't understand the difference between factors and multiples. Can you explain it in a simple way?

Answer: Great question! Think of it this way:

  • Factors are numbers that GO INTO another number (like 3 goes into 12 three times)
  • Multiples are numbers that come OUT of multiplying the original number (like 12 comes out of 3×4)

Think of it like a factory: factors are the ingredients going IN, multiples are the products coming OUT.

Example with number 6:

  • Factors of 6: 1, 2, 3, 6 (these go into 6 evenly)
  • Multiples of 6: 6, 12, 18, 24, 30... (these come from 6×1, 6×2, 6×3...)

Remember: FACTORS are usually smaller than the number, MULTIPLES are usually bigger.

Question: Why do we need to find the LCM? When would we use it in real life?

Answer: LCM has many practical uses! Here are some real-life examples:

  • Event scheduling: If bus A comes every 6 minutes and bus B every 8 minutes, they'll meet every 24 minutes (the LCM of 6 and 8)
  • Cooking: If one recipe serves 3 people and another serves 4, you'd need the LCM (12) to make enough for both recipes
  • Shopping: If packs of pencils come in sets of 6 and erasers in sets of 8, you'd buy LCM amount (24) to have matching quantities

In math, LCM helps us add fractions with different denominators by finding a common denominator.

It's about finding the smallest common meeting point between different repeating cycles!

Question: How do I quickly identify if a number is prime without checking every number?

Answer: There are several shortcuts to test if a number is prime:

  • Even numbers: Any even number greater than 2 is composite (divisible by 2)
  • Divisibility by 3: If the sum of digits is divisible by 3, the number is composite
  • Ends in 5: Any number ending in 5 (except 5 itself) is composite
  • Square root rule: Only test divisors up to the square root of the number

For example, to test 37:

  • Not even, doesn't end in 5
  • Sum of digits: 3+7=10 (not divisible by 3)
  • Test only up to √37 ≈ 6.1: try 2, 3, 5
  • None work, so 37 is prime

Memorize primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47