Solved Exercises on Understanding Factors in Grade 4

Master factors: identifying factors, finding all factors, prime numbers, and factor pairs through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Factors
Exercise 1
Which of these numbers are factors of 12: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12? List all the factors of 12.
Definition:

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

Finding Factors Method:
  1. Test each number: Divide the target number by each potential factor
  2. Check remainder: If remainder is 0, it's a factor
  3. List systematically: Go from 1 to the target number
  4. Stop early: When factors start repeating, you're done
Target Number
12
Potential Factors
1-12
Actual Factors
?
Step 1: Test each number from 1 to 12

12 ÷ 1 = 12 R0 → 1 is a factor
12 ÷ 2 = 6 R0 → 2 is a factor
12 ÷ 3 = 4 R0 → 3 is a factor
12 ÷ 4 = 3 R0 → 4 is a factor
12 ÷ 5 = 2 R2 → 5 is NOT a factor
12 ÷ 6 = 2 R0 → 6 is a factor
12 ÷ 7 = 1 R5 → 7 is NOT a factor
12 ÷ 8 = 1 R4 → 8 is NOT a factor
12 ÷ 9 = 1 R3 → 9 is NOT a factor
12 ÷ 10 = 1 R2 → 10 is NOT a factor
12 ÷ 11 = 1 R1 → 11 is NOT a factor
12 ÷ 12 = 1 R0 → 12 is a factor

Step 2: List all factors found

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

Step 3: Verify using multiplication pairs

1 × 12 = 12 ✓
2 × 6 = 12 ✓
3 × 4 = 12 ✓

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

The factors of 12 are: 1, 2, 3, 4, 6, 12. From the given list, these numbers divide 12 evenly with no remainder.

Applied rules:

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

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

• Systematic approach: Test numbers in order to avoid missing any factors

2 Finding All Factors
Exercise 2
Find all the factors of 18. Show your work by listing factor pairs.
Definition:

Factor pairs: Two numbers that multiply together to give the target number. For example, 2 and 9 are a factor pair of 18 because 2 × 9 = 18.

Target Number
18
Factor Pairs
?
All Factors
?
Step 1: Find factor pairs systematically

Start with 1 and work up:
1 × 18 = 18 → (1, 18)
2 × 9 = 18 → (2, 9)
3 × 6 = 18 → (3, 6)
4 × ? = 18 → 18 ÷ 4 = 4.5 (not a whole number, so not a factor)
5 × ? = 18 → 18 ÷ 5 = 3.6 (not a whole number, so not a factor)
6 × 3 = 18 → Already found (3, 6)
Stop here since we've reached a factor we already listed.

Step 2: List all unique factors

From the pairs (1, 18), (2, 9), (3, 6), we get: 1, 2, 3, 6, 9, 18

Step 3: Verify all factors

18 ÷ 1 = 18 R0 ✓
18 ÷ 2 = 9 R0 ✓
18 ÷ 3 = 6 R0 ✓
18 ÷ 6 = 3 R0 ✓
18 ÷ 9 = 2 R0 ✓
18 ÷ 18 = 1 R0 ✓

Step 4: Arrange in order

Factors of 18 in ascending order: 1, 2, 3, 6, 9, 18

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

All factors of 18 are: 1, 2, 3, 6, 9, 18. These can be organized as factor pairs: (1, 18), (2, 9), (3, 6).

Applied rules:

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

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

• Order: List factors in ascending order for clarity

3 Prime Numbers
Exercise 3
Which of these numbers are prime: 7, 8, 9, 11, 13? Explain your reasoning.
Definition:

Prime number: A number that has exactly two factors: 1 and itself. Composite number: A number that has more than two factors.

Numbers to Test
7, 8, 9, 11, 13
Prime Status
?
Reasoning
?
Step 1: Test number 7

Factors of 7: 1, 7
Only two factors → 7 is prime

Step 2: Test number 8

Factors of 8: 1, 2, 4, 8
More than two factors → 8 is composite

Step 3: Test number 9

Factors of 9: 1, 3, 9
More than two factors → 9 is composite

Step 4: Test number 11

Factors of 11: 1, 11
Only two factors → 11 is prime

Step 5: Test number 13

Factors of 13: 1, 13
Only two factors → 13 is prime

Step 6: Compile the answer

Prime numbers have exactly two factors: 1 and themselves

Prime: 7, 11, 13
Final answer:

The prime numbers are: 7, 11, 13. These numbers each have exactly two factors: 1 and the number itself. The composite numbers are 8 and 9, which have more than two factors.

Applied rules:

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

• Composite definition: More than two factors

• Testing method: Find all factors and count them

Factors: Definitions, Properties, and Methods
a × b = c, so a and b are factors of c
Factor Definition
Factor Pair
2 × 6 = 12
So 2 and 6 are factors of 12
Prime Number
7 has factors: 1, 7
Exactly 2 factors → Prime
Composite Number
8 has factors: 1, 2, 4, 8
More than 2 factors → Composite
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 number: A number with exactly two factors: 1 and itself.

Composite number: A number with more than two factors.

Multiple: The product of a number and any whole number.

Finding factors methods:
  1. Division method: Divide the target number by each number from 1 to the target
  2. Multiplication pairs: Find pairs of numbers that multiply to the target
  3. Factor trees: Break down the number into prime factors
  4. Systematic listing: Work in order to avoid missing factors
Tip 1: 1 and the number itself are always factors of any number.
Tip 2: Stop checking when you reach a factor you already paired with another.
Tip 3: Even numbers (except 2) are never prime since they're divisible by 2.
Tip 4: Once you find a factor, its pair is the result of dividing the target by that factor.
Factor properties: Every number has at least two factors (1 and itself). Factors come in pairs that multiply to the original number.
Prime numbers: 2 is the only even prime number. All other primes are odd. There are infinitely many prime numbers.
Important factor 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 (except for 0)

• Prime numbers have exactly two factors

• Composite numbers have more than two factors

Factor Patterns and Relationships
Factor Count Analysis
Visualizing the number of factors for different numbers to identify patterns.

Analysis: The chart shows how the number of factors varies across different numbers.

  • Prime numbers have exactly 2 factors
  • Square numbers have an odd number of factors
  • Highly composite numbers have many factors
Solution: Exercises 4 to 5
4 Factor Applications
Exercise 4
A rectangular garden has an area of 24 square feet. What are all the possible dimensions (length and width) of the garden? Explain how factors help solve this problem.
Definition:

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

Area of Garden
24 sq ft
Length × Width
= 24
Possible Dimensions
?
Step 1: Identify the connection to factors

Since Area = Length × Width = 24, we need to find all factor pairs of 24

Step 2: Find all factor pairs of 24

1 × 24 = 24 → (1, 24)
2 × 12 = 24 → (2, 12)
3 × 8 = 24 → (3, 8)
4 × 6 = 24 → (4, 6)
6 × 4 = 24 → Already found (4, 6)
Stop here since we've reached a factor we already paired.

Step 3: List all possible dimensions

Possible dimensions (length, width):
(1, 24), (2, 12), (3, 8), (4, 6)
Or, considering orientation doesn't matter: 1×24, 2×12, 3×8, 4×6

Step 4: Verify each combination

1 × 24 = 24 ✓
2 × 12 = 24 ✓
3 × 8 = 24 ✓
4 × 6 = 24 ✓

Step 5: Explain the connection

Factors help because the length and width must multiply to give the area. Finding factor pairs directly gives us the possible dimensions.

Dimensions: 1×24, 2×12, 3×8, 4×6
Final answer:

The possible dimensions are: 1×24, 2×12, 3×8, or 4×6 feet. Since Area = Length × Width, we needed to find all factor pairs of 24, which represent all possible length and width combinations.

Applied rules:

• Area formula: Length × Width = Area

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

• Practical consideration: Both orientations represent the same rectangle

5 Advanced Factor Patterns
Exercise 5
Look at the factors of 16: 1, 2, 4, 8, 16. Notice that 4 appears only once in the list, even though 4 × 4 = 16. Explain why this happens and identify other numbers with this property.
Definition:

Square numbers: Numbers that can be expressed as a number multiplied by itself (n²). These have an odd number of factors because one factor is repeated.

Square Number
16 = 4²
Factors of 16
1, 2, 4, 8, 16
Factor Pairs
(1,16), (2,8), (4,4)
Step 1: Examine factor pairs of 16

1 × 16 = 16 → (1, 16)
2 × 8 = 16 → (2, 8)
4 × 4 = 16 → (4, 4)

Step 2: Analyze the special case

In the pair (4, 4), the same number appears twice. When listing factors, we only write 4 once, not twice.

Step 3: Identify the pattern

This happens with all square numbers because they have one factor that multiplies by itself to give the original number.

Step 4: Identify other square numbers

Other examples: 1 (1²), 4 (2²), 9 (3²), 25 (5²), 36 (6²), 49 (7²), 64 (8²), 81 (9²), 100 (10²)

Step 5: Note the pattern in factor counts

Square numbers have an odd number of factors, while non-square numbers have an even number of factors.

Step 6: Verify with another example

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

Square numbers: 1, 4, 9, 16, 25, 36, 49...
Final answer:

This happens because 16 is a square number (4²). In the factor pair (4, 4), the same number appears twice, but we only list it once in the factors list. Other square numbers like 1, 4, 9, 25, 36, 49, etc., have this same property where one factor is repeated, resulting in an odd total count of factors.

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

Complete Summary: Understanding Factors
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 number: A number with exactly two factors: 1 and itself.

Composite number: A number with more than two factors.

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

Complete methodology:
  1. Identify the target number: Know what number you're finding factors for.
  2. Systematically test numbers: Divide the target by each number from 1 up to the target.
  3. Record factors: When remainder is 0, record the divisor as a factor.
  4. Use factor pairs: For each factor found, calculate its pair by dividing the target by that factor.
  5. Stop appropriately: When you reach a factor you've already paired, stop searching.
  6. Organize results: List factors in ascending order.
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.
Key patterns: Prime numbers have exactly 2 factors, composite numbers have more than 2, square numbers have odd factor counts.
Factor 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 (except for 0)

• Prime numbers have exactly two factors

• Composite numbers have more than two factors

• Square numbers have an odd number of factors

Exercise with Visualization: Factor Trees
Exercise 6: Prime Factorization
Visualizing how numbers can be broken down into prime factors:
f₁(x) = Prime factors of x (for x = 12, 18, 20, 24)
f₂(x) = Number of prime factors (for x = 12, 18, 20, 24)
f₃(x) = Unique prime factors (for x = 12, 18, 20, 24)

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 is 1 not considered a prime number? It only has one factor, which is itself.

Answer: This is a great question! Although 1 only has one factor (itself), mathematicians define prime numbers as having exactly two factors: 1 and the number itself. Since 1 only has one factor total, it doesn't meet this definition.

Additionally, if 1 were considered prime, it would cause problems in mathematical theorems. For example, the fundamental theorem of arithmetic states that every number can be uniquely expressed as a product of primes. If 1 were prime, we could write 6 as 2×3 or 1×2×3 or 1×1×2×3, etc., destroying the uniqueness.

Question: How do I know when to stop looking for factors? Sometimes I feel like I might miss some.

Answer: There are two reliable ways to know when you're done:

  • Method 1: When you reach a factor that you've already paired with another factor, you're done. For example, with 12: when you check 4, you get 12÷4=3, and you already have the pair (3,4).
  • Method 2: Stop when the quotient becomes smaller than the divisor. For 12: when checking 4, you get 3; when checking 5, you get 2.4 (not a factor anyway).

Working systematically from 1 upward helps ensure you don't miss any factors.

Question: My child is struggling to understand factors. Are there any visual methods to help them grasp the concept?

Answer: Yes! Visual methods are very effective for understanding factors:

  • Array models: Draw rectangles with rows and columns matching factor pairs (e.g., for 12: 3×4 array)
  • Factor rainbows: Draw arcs connecting factor pairs in order
  • Manipulatives: Use blocks or counters to physically group numbers
  • Factor trees: Break numbers down into prime factors visually
  • Grid paper: Shade rectangles to represent different factor pairs

These concrete representations help students see the multiplicative relationships.

Question: Why do square numbers have an odd number of factors? It seems weird that 16 has 5 factors instead of an even number.

Answer: This happens because one of the factors is repeated! For 16, the factor pairs are:

  • (1, 16) - two different factors
  • (2, 8) - two different factors
  • (4, 4) - one factor that appears twice in the multiplication

When we list all factors of 16: 1, 2, 4, 8, 16, the 4 only appears once in the list, even though it was used twice in the multiplication (4×4). For non-square numbers, all factors come in distinct pairs, giving an even total. For square numbers, one factor is "paired" with itself, creating an odd total.

Question: What's the difference between factors and multiples? Sometimes I mix them up.

Answer: Here's how to remember the difference:

  • Factors: Numbers that divide INTO a number evenly. Factors are always smaller than or equal to the number. Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
  • Multiples: Numbers you get BY multiplying the original number. Multiples are always equal to or larger than the number. Example: Multiples of 3 are 3, 6, 9, 12, 15, 18...

Think: "Factors are few" (limited number of factors), "Multiples are many" (infinite number of multiples).