Solved Exercises on Finding Factors of Numbers in Grade 4

Master finding factors: systematic methods, factor trees, and factor patterns through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Systematic Method
Exercise 1
Find all factors of 24 using the systematic division method. Show your work.
Definition:

Systematic division method: A method of finding factors by dividing the target number by each integer from 1 up to the number, recording divisors that result in no remainder.

Systematic Finding Factors Method:
  1. Start with 1: Every number is divisible by 1
  2. Test sequentially: Divide by 2, 3, 4, etc.
  3. Record factors: When remainder is 0, record the divisor
  4. Stop efficiently: When you reach a factor you already paired
  5. Organize: List factors in ascending order
Target Number
24
Factors Found
?
Method
Systematic
Step 1: Start with 1

24 ÷ 1 = 24 R0 → 1 and 24 are factors

Step 2: Test 2

24 ÷ 2 = 12 R0 → 2 and 12 are factors

Step 3: Test 3

24 ÷ 3 = 8 R0 → 3 and 8 are factors

Step 4: Test 4

24 ÷ 4 = 6 R0 → 4 and 6 are factors

Step 5: Test 5

24 ÷ 5 = 4 R4 → 5 is NOT a factor

Step 6: Test 6

24 ÷ 6 = 4 R0 → 6 is a factor, but we already found the pair (4,6)

Step 7: Stop here

Since we've reached a factor (6) that we already paired with 4, we're done.

Step 8: List all factors

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

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

All factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24. These can be organized as factor pairs: (1,24), (2,12), (3,8), (4,6).

Applied rules:

• Factor definition: A number is a factor if it divides the target with remainder 0

• Factor pairs: Every factor has a partner that multiplies to the target

• Stopping point: When you reach a factor you already paired, stop searching

2 Factor Tree Method
Exercise 2
Create a factor tree for 36 and use it to find all factors of 36.
Definition:

Factor tree: A visual representation of how a number can be broken down into its prime factors through repeated division.

Target Number
36
Factor Tree
?
All Factors
?
Step 1: Start the factor tree

36 can be split into 4 × 9
36
/ \
4 9

Step 2: Continue breaking down

4 = 2 × 2 and 9 = 3 × 3
36
/ \
4 9
/ \ / \
2 2 3 3

Step 3: Write prime factorization

36 = 2² × 3²

Step 4: Find all factors using prime factorization

To find all factors, we take all possible combinations of the prime factors:
2⁰ × 3⁰ = 1
2¹ × 3⁰ = 2
2² × 3⁰ = 4
2⁰ × 3¹ = 3
2¹ × 3¹ = 6
2² × 3¹ = 12
2⁰ × 3² = 9
2¹ × 3² = 18
2² × 3² = 36

Step 5: List all factors

Factors of 36 in order: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 6: Verify factor pairs

(1,36), (2,18), (3,12), (4,9), (6,6)

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Final answer:

Using the factor tree, we determined that 36 = 2² × 3². All factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Applied rules:

• Factor trees: Help visualize how numbers break down into prime factors

• Prime factorization: Unique representation of a number as a product of primes

• Factor generation: All factors can be found using combinations of prime factors

3 Pattern Recognition
Exercise 3
Find all factors of 16, 25, and 36. What pattern do you notice about these numbers and their factors?
Definition:

Square numbers: Numbers that can be expressed as n², where n is an integer. These have an interesting property regarding their factors.

Square Numbers
16, 25, 36
Their Factors
?
Pattern
?
Step 1: Find factors of 16

16 = 4²
Factors of 16: 1, 2, 4, 8, 16 (count: 5 factors)

Step 2: Find factors of 25

25 = 5²
Factors of 25: 1, 5, 25 (count: 3 factors)

Step 3: Find factors of 36

36 = 6²
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (count: 9 factors)

Step 4: Analyze factor pairs

16: (1,16), (2,8), (4,4) - Note that 4 is paired with itself
25: (1,25), (5,5) - Note that 5 is paired with itself
36: (1,36), (2,18), (3,12), (4,9), (6,6) - Note that 6 is paired with itself

Step 5: Identify the pattern

For square numbers, one factor is repeated (the square root), which results in an odd total number of factors.

Step 6: Generalize the pattern

Non-square numbers have factors that all come in distinct pairs, giving an even count. Square numbers have one factor that pairs with itself, giving an odd count.

Square numbers: Odd number of factors
Final answer:

The pattern is that all three numbers (16, 25, 36) are square numbers, and they all have an odd number of factors. This is because in square numbers, the square root appears only once in the factor list, even though it's used twice in multiplication.

Applied rules:

• Square number property: Has a repeated factor (n × n = n²)

• Odd factor count: Square numbers have an odd number of factors

• Even factor count: Non-square numbers have an even number of factors

Finding Factors: Methods, Properties, and Patterns
a × b = c, so a and b are factors of c
Factor Definition
Systematic Method
Test 1 to n
Divide and check remainder
Factor Tree
Break into primes
Build from prime factors
Square Numbers
n² has odd factors
√n appears once
Key definitions:

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

Factor pair: Two numbers that multiply together to give a product.

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

Square number: A number that can be expressed as n².

Factor tree: A visual representation showing how a number breaks down into prime factors.

Finding factors methods:
  1. Systematic division: Divide by each number from 1 to the target
  2. Factor pairs: Find pairs that multiply to the target
  3. Factor trees: Break down into prime factors
  4. Pattern recognition: Use known patterns for special numbers
Tip 1: Always start with 1 and the number itself as factors.
Tip 2: Stop checking when you reach a factor you already paired with another.
Tip 3: Use factor trees to understand the structure of numbers.
Tip 4: Square numbers have an odd number of factors because one factor is repeated.
Factor properties: Every number has at least two factors (1 and itself). Factors come in pairs that multiply to the original number.
Pattern properties: Prime numbers have exactly 2 factors, composite numbers have more than 2, square numbers have odd factor counts.
Important finding factors rules:

• Every number is a factor of itself

• 1 is a factor of every number

• Factors are always less than or equal to the number

• Factor pairs multiply to give the original number

• Square numbers have an odd number of factors

Factor Finding Methods Comparison
Method Efficiency
Visualizing the efficiency of different methods for finding factors.

Analysis: The chart shows how different methods compare for various number sizes.

  • Systematic method works well for small numbers
  • Factor trees are helpful for understanding structure
  • Pattern recognition speeds up the process
Solution: Exercises 4 to 5
4 Advanced Pattern Recognition
Exercise 4
Find all factors of 60. Then group them into sets that multiply to 60. What patterns do you notice about these factor sets?
Definition:

Factor sets: Groups of factors that multiply together to give the original number. For example, for 60: 4 × 15 = 60, or 2 × 6 × 5 = 60.

Target Number
60
All Factors
?
Factor Sets
?
Step 1: Find all factors of 60

Using systematic method:
60 ÷ 1 = 60 R0 → 1, 60
60 ÷ 2 = 30 R0 → 2, 30
60 ÷ 3 = 20 R0 → 3, 20
60 ÷ 4 = 15 R0 → 4, 15
60 ÷ 5 = 12 R0 → 5, 12
60 ÷ 6 = 10 R0 → 6, 10
60 ÷ 7 = 8 R4 → Not a factor
60 ÷ 8 = 7 R4 → Not a factor
60 ÷ 9 = 6 R6 → Not a factor
60 ÷ 10 = 6 R0 → Already found (6,10)
Stop here.

Step 2: List all factors

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Step 3: Group into factor pairs

(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)

Step 4: Explore factor sets with more than 2 factors

Using prime factorization: 60 = 2² × 3 × 5 = 2 × 2 × 3 × 5
Some sets: {2, 2, 3, 5}, {2, 30}, {3, 20}, {4, 15}, {5, 12}, {6, 10}

Step 5: Identify patterns

1. Each factor pair multiplies to 60
2. The product of any factor and its "complement" equals 60
3. Prime factorization reveals the fundamental building blocks

Step 6: Count factor pairs

There are 12 factors total, forming 6 factor pairs. For any number, the total number of factors is even unless it's a perfect square.

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Final answer:

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. These form 6 factor pairs: (1,60), (2,30), (3,20), (4,15), (5,12), (6,10). The pattern shows that factors come in pairs whose product is the original number.

Applied rules:

• Factor pairs: Every factor has a complement that multiplies to the original number

• Prime factorization: Reveals the fundamental structure of numbers

• Pairing principle: Factors naturally group into pairs for non-square numbers

5 Problem-Solving Application
Exercise 5
A rectangular room has an area of 48 square feet. What are all the possible dimensions of the room? If the room must be at least 6 feet wide, which dimensions are valid?
Definition:

Area application: Problems where the area of a rectangle corresponds to finding factor pairs of that area, representing possible length and width combinations.

Area of Room
48 sq ft
Constraint
≥ 6 ft wide
Valid Dimensions
?
Step 1: Find all factor pairs of 48

Using systematic method:
48 ÷ 1 = 48 R0 → (1, 48)
48 ÷ 2 = 24 R0 → (2, 24)
48 ÷ 3 = 16 R0 → (3, 16)
48 ÷ 4 = 12 R0 → (4, 12)
48 ÷ 5 = 9 R3 → Not a factor
48 ÷ 6 = 8 R0 → (6, 8)
48 ÷ 7 = 6 R6 → Not a factor
48 ÷ 8 = 6 R0 → Already found (6, 8)
Stop here.

Step 2: List all possible dimensions

Without constraints: (1, 48), (2, 24), (3, 16), (4, 12), (6, 8)

Step 3: Apply the constraint (at least 6 feet wide)

We need dimensions where at least one measurement is ≥ 6:
(1, 48) → Invalid (1 < 6)
(2, 24) → Invalid (2 < 6)
(3, 16) → Invalid (3 < 6)
(4, 12) → Invalid (4 < 6)
(6, 8) → Valid (both ≥ 6)

Step 4: Reconsider the constraint

Actually, if "at least 6 feet wide" means the width must be ≥ 6, then valid dimensions are those where at least one dimension is ≥ 6:
(6, 8) → Width 6, Length 8 OR Width 8, Length 6 (both valid)

Step 5: Clarify the constraint

If the width specifically must be ≥ 6, then valid dimensions are: 6×8 feet, 8×6 feet, 12×4 feet, 16×3 feet, 24×2 feet, 48×1 feet (but only if width ≥ 6)

Step 6: Interpret the constraint

Assuming "at least 6 feet wide" means at least one dimension is ≥ 6, then all factor pairs except (1,48) and (2,24) where both dimensions are < 6 are valid. Actually, (3,16) and (4,12) have one dimension ≥ 6, so they're valid too.

Valid dimensions: (6,8), (8,6), (12,4), (16,3), (24,2), (48,1)
Final answer:

For a 48 sq ft room with at least one dimension ≥ 6 ft, the valid dimensions are: 6×8, 8×6, 12×4, 16×3, 24×2, and 48×1 feet. These correspond to the factor pairs where at least one factor is 6 or greater.

Applied rules:

• Area formula: Length × Width = Area

• Factor connection: Dimension pairs are factor pairs of the area

• Constraint application: Filter solutions based on problem requirements

Complete Summary: Finding Factors of Numbers
a × b = c, so a and b are factors of c
Factor Definition
Key definitions:

Factor: A number that divides another number evenly with no remainder. If a × b = c, then both a and b are factors of c.

Factor pair: Two numbers that multiply together to give a product.

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

Square number: A number that can be expressed as n², having a repeated factor.

Factor tree: A visual representation showing how a number breaks down into prime factors.

Complete methodology:
  1. Identify the target number: Know what number you're finding factors for.
  2. Choose a method: Systematic division, factor pairs, or factor trees.
  3. Systematically find factors: Divide the target by each number from 1 up to the target.
  4. Record factors: When remainder is 0, record the divisor as a factor.
  5. Use factor pairs: For each factor found, calculate its pair by dividing the target by that factor.
  6. Stop appropriately: When you reach a factor you've already paired, stop searching.
  7. Organize results: List factors in ascending order.
  8. Verify completeness: Ensure all factor pairs multiply to the target.
Tip 1: Always remember that 1 and the number itself are factors of any number.
Tip 2: Work systematically from smallest to largest to avoid missing factors.
Tip 3: When you find a factor, its pair is simply the target divided by that factor.
Tip 4: Square numbers have an odd number of factors because one factor is repeated.
Common errors: Forgetting 1 and the number itself as factors, missing factor pairs, counting repeated factors twice in square numbers, stopping too early.
Key patterns: Prime numbers have exactly 2 factors, composite numbers have more than 2, square numbers have odd factor counts, factor pairs multiply to the original number.
Finding factors rules:

• Every number is a factor of itself

• 1 is a factor of every number

• Factors are always less than or equal to the number

• Factor pairs multiply to give the original number

• Prime numbers have exactly two factors

• Square numbers have an odd number of factors

Exercise with Visualization: Factor Trees
Exercise 6: Prime Factorization Trees
Visualizing how numbers decompose into prime factors:
f₁(x) = Prime factors of x (for x = 24, 36, 48, 60)
f₂(x) = Number of prime factors (for x = 24, 36, 48, 60)
f₃(x) = Unique prime factors (for x = 24, 36, 48, 60)

Analysis: The chart shows how numbers decompose into prime factors.

  • Every composite number can be expressed as a product of primes
  • Prime factorization is unique for each number
  • Prime factors help understand divisibility and multiples

Questions & Answers

Question: Why do we stop looking for factors when we reach a factor we already paired? How do I know I won't miss any?

Answer: This happens because factors come in pairs! Let's say you're finding factors of 24:

  • When you check 4, you find that 24 ÷ 4 = 6, so (4, 6) is a factor pair
  • Later, when you check 6, you find that 24 ÷ 6 = 4, so (6, 4) is the same pair
  • Since you already found the pair (4, 6), there's no need to record (6, 4) separately

This means you've found all possible factor pairs and can stop searching.

Question: Which method is best for finding factors: systematic division or factor trees? I'm confused about when to use each.

Answer: Both methods are useful for different purposes:

  • Systematic division: Best for quickly listing ALL factors in order. Good for small to medium numbers.
  • Factor trees: Best for understanding the structure of numbers and finding prime factorization. Helpful for larger numbers or when you need to understand divisibility.

For most grade 4 problems, systematic division is more direct for finding all factors.

Question: My child is having trouble with factor trees. Are there any tricks to make them easier to understand?

Answer: Yes! Try these strategies:

  • Start with easy splits: Look for obvious factors like 2, 3, 5, 10 first
  • Think of multiplication facts: "What times what equals this number?"
  • Draw branches clearly: Each split creates two branches
  • Circle primes: Once you reach a prime number, circle it and stop
  • Check your work: Multiply all the circled primes to ensure you get the original number

Practice with smaller numbers first before moving to larger ones.

Question: Why do square numbers have an odd number of factors? It still doesn't make sense to me.

Answer: Think of it this way: factors usually come in pairs that multiply to the original number. For example, with 12: (1,12), (2,6), (3,4). That's 6 factors total (even number).

But with square numbers, one factor multiplies by itself: √n × √n = n. For 16: (1,16), (2,8), (4,4). The 4 only appears once in the list of factors, even though it was used twice in multiplication. So we have 5 factors total (odd number).

The square root "goes alone" instead of pairing with a different number!

Question: How can I check if I've found all the factors of a number? Sometimes I'm not sure if I missed any.

Answer: Here are two reliable checks:

  • Pairing check: Make sure every factor has a "partner" that multiplies to the original number. For 24: 1×24=24, 2×12=24, 3×8=24, 4×6=24
  • Systematic check: Go back and try dividing the original number by each number from 1 to the original number to verify you didn't miss any
  • Stopping rule: When you reach a factor you already paired with another, you're done

If you follow the systematic method correctly, you won't miss any factors.