Solved Exercises on Finding Factors of Numbers in Grade 4

Master finding factors of numbers: definition, divisibility rules, factor trees, and practice problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Factor Identification
Exercise 1
Find all factors of 12.
Visual: 🧮 12 ÷ ? = whole number
Definition:

Factor: A number that divides another number evenly without remainder

Finding Factors Method:
  1. Start with 1 and the number itself
  2. Test consecutive numbers up to half the number
  3. If division results in a whole number, both divisors are factors
  4. List all factors in ascending order
Number
12
Factors
1, 2, 3, 4, 6, 12
Step 1: Start with 1 and 12

12 ÷ 1 = 12 → 1 and 12 are factors

Step 2: Test 2

12 ÷ 2 = 6 → 2 and 6 are factors

Step 3: Test 3

12 ÷ 3 = 4 → 3 and 4 are factors

Step 4: Test 4

We already found 4, so stop here

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

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

Applied rules:

• Division Test: If n ÷ d = whole number, then d is a factor of n

• Pairing: Every factor has a pair (d × ? = n)

• Stopping Rule: Stop when you reach √n or find repeated factors

2 Factor Tree Method
Exercise 2
Find all factors of 18 using factor tree method.
Visual: 🌳 18 = 2 × 9 = 2 × 3 × 3
Definition:

Factor Tree: A diagram showing how a number breaks down into its prime factors

Number
18
Prime Factors
2 × 3²
All Factors
1, 2, 3, 6, 9, 18
Step 1: Draw factor tree for 18

18
├─ 2
└─ 9
├─ 3
└─ 3

Step 2: List all combinations

1 (always a factor)
2
3
2×3=6
3×3=9
2×3×3=18

Step 3: Arrange in order

1, 2, 3, 6, 9, 18

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

The factors of 18 are: 1, 2, 3, 6, 9, 18

Applied rules:

• Factor Tree: Break down into prime factors

• Combination: Create all possible products of prime factors

• Order: List factors in ascending order

3 Divisibility Rules Application
Exercise 3
Find all factors of 24 using divisibility rules.
Visual: 🔢 24 ÷ 2, 3, 4, 6, 8, 12 = whole numbers
Definition:

Divisibility Rules: Quick tests to determine if one number divides another

Number
24
Divisibility Tests
2, 3, 4, 6, 8
All Factors
1, 2, 3, 4, 6, 8, 12, 24
Step 1: Apply divisibility rules

• 2: Last digit is even (4) → 24÷2=12
• 3: Sum of digits (2+4=6) divisible by 3 → 24÷3=8
• 4: Last two digits (24) divisible by 4 → 24÷4=6

Step 2: Find remaining factors

Test 5: 24÷5=4.8 (not whole)
Test 6: 24÷6=4 (already found)
Stop at √24≈5

Step 3: List all factors

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:

• Divisibility by 2: Even last digit

• Divisibility by 3: Sum of digits divisible by 3

• Divisibility by 4: Last two digits divisible by 4

Rules and methods, laws,...
a × b = n → a and b are factors of n
Factor Relationship
Rule 1
1 × n = n
1 and n are always factors
Rule 2
a × b = n
Both a and b are factors
Rule 3
√n stopping point
Stop when reaching √n
Key definitions:

Factor: A number that divides another number evenly

Multiple: A number that results from multiplying a factor by an integer

Prime Factor: A factor that is also a prime number

Complete methodology:
  1. Identify the number: Determine which number to find factors of
  2. Start with 1: Always include 1 and the number itself
  3. Test systematically: Check numbers 2, 3, 4, etc. up to √n
  4. Record pairs: For each factor found, record both numbers
  5. Verify completeness: Ensure all factors are listed
Tip 1: Always include 1 and the number itself as factors.
Tip 2: Only test numbers up to the square root of the original number.
Tip 3: Factors come in pairs: if a is a factor, then n/a is also a factor.
Tip 4: Use divisibility rules to speed up the process.
Common errors: Forgetting 1 and the number itself, missing factor pairs, going beyond √n unnecessarily.
Exam preparation: Practice with numbers 1-50, memorize divisibility rules, check your work systematically.
Formulas to know by heart:

• If a × b = n, then both a and b are factors of n

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

• Prime numbers have exactly two factors: 1 and themselves

• Composite numbers have more than two factors

Solution: Exercises 4 to 5
4 Complex Factor Analysis
Exercise 4
Find all factors of 36 and identify which are prime numbers.
Visual: 🔍 36 = 2² × 3² → factors from combinations
Definition:

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

Number
36
Prime Factorization
2² × 3²
All Factors
1, 2, 3, 4, 6, 9, 12, 18, 36
Step 1: Find factor pairs systematically

36 ÷ 1 = 36 → (1, 36)
36 ÷ 2 = 18 → (2, 18)
36 ÷ 3 = 12 → (3, 12)
36 ÷ 4 = 9 → (4, 9)
36 ÷ 6 = 6 → (6, 6) - stop here

Step 2: List all unique factors

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

Step 3: Identify prime factors

Prime numbers in the list: 2 and 3

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

All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Prime factors: 2, 3

Applied rules:

• Systematic Testing: Check divisors up to √36 = 6

• Prime Identification: Check if each factor is prime

• Verification: Ensure all factor pairs multiply to original number

5 Problem Solving with Factors
Exercise 5
Sarah has 20 cookies and wants to arrange them in equal rows. In how many different ways can she arrange them?
Visual: 🍪 Cookies arranged in rectangles with different dimensions
Definition:

Equal Rows: Arranging items in rectangular formation where number of rows × number of columns = total items

Problem
20 cookies in equal rows
Find
Factors of 20
Arrangements
1×20, 2×10, 4×5, 5×4, 10×2, 20×1
Step 1: Find all factors of 20

20 ÷ 1 = 20 → (1, 20)
20 ÷ 2 = 10 → (2, 10)
20 ÷ 4 = 5 → (4, 5)
20 ÷ 5 = 4 → already found
Stop at √20 ≈ 4.5

Step 2: Interpret as arrangements

1 row of 20 cookies
2 rows of 10 cookies
4 rows of 5 cookies
5 rows of 4 cookies
10 rows of 2 cookies
20 rows of 1 cookie

Step 3: Count unique arrangements

There are 6 different arrangements possible

Factors of 20: 1, 2, 4, 5, 10, 20
Number of arrangements: 6
Final answer:

Sarah can arrange 20 cookies in 6 different ways:
1×20, 2×10, 4×5, 5×4, 10×2, 20×1

Applied rules:

• Factor Pairs: Each factor pair represents a rectangle dimension

• Real-world Application: Factors solve practical arrangement problems

• Verification: Each arrangement equals total number of items

Complete Guide: Laws, Methods, Rules, Definitions
a × b = n ↔ a|n ∧ b|n
Factor Relationship
Key definitions:

Factor: A number that divides another number evenly without remainder

Multiple: A number that results from multiplying a factor by an integer

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

Composite Number: A number with more than two factors

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

Complete methodology:
  1. Identify the target number for which you need to find factors
  2. Start with 1 and the number itself (always factors)
  3. Test consecutive integers starting from 2 up to √n
  4. For each divisor found, record both the divisor and quotient
  5. Stop when reaching √n to avoid duplicates
  6. List factors in ascending order for clarity
Tip 1: Use divisibility rules to quickly eliminate non-factors.
Tip 2: Remember that factors come in pairs: if a is a factor, then n/a is also a factor.
Tip 3: When you find a factor pair (a, b) where a = b, you've reached √n and can stop.
Tip 4: Always verify by checking that each factor pair multiplies back to the original number.
Divisibility Rules: 2 (even), 3 (sum of digits), 4 (last two digits), 5 (ends in 0 or 5), 6 (divisible by 2 and 3), 9 (sum of digits), 10 (ends in 0).
Prime vs Composite: Prime numbers have exactly 2 factors (1 and themselves), composite numbers have more than 2 factors.
Essential Rules to Remember:

• Every number is divisible by 1 and itself

• The largest factor (other than the number itself) is at most n/2

• Factors come in pairs: a × b = n means both a and b are factors

• If a number has exactly two factors, it's prime

• The number of factors can be determined from prime factorization

Visualization: Factors of Different Numbers
Comparing Factors
Compare the number of factors for: 12, 16, 18, 20, 24
Visual: 📊 More factors indicate more divisibility options

Analysis: The chart shows how different numbers have varying numbers of factors.

  • 12: 1, 2, 3, 4, 6, 12 → 6 factors
  • 16: 1, 2, 4, 8, 16 → 5 factors
  • 18: 1, 2, 3, 6, 9, 18 → 6 factors
  • 20: 1, 2, 4, 5, 10, 20 → 6 factors
  • 24: 1, 2, 3, 4, 6, 8, 12, 24 → 8 factors

Questions & Answers

Question: I don't understand why we only need to check up to the square root of a number when finding factors. Why don't we check all the way up to the number itself?

Answer: Great question! This is because factors come in pairs. When you find one factor, you automatically find its pair.

For example, with 12:

  • When you find 2 as a factor (12 ÷ 2 = 6), you also automatically find 6 as a factor
  • When you find 3 as a factor (12 ÷ 3 = 4), you also automatically find 4 as a factor

Once you reach √12 ≈ 3.5, you've already found all the smaller factors and their pairs. Any number larger than √12 would have already appeared as a factor in a previous pair. This makes finding factors much more efficient!

Question: What's the difference between a factor and a multiple? They seem very similar to me.

Answer: Factors and multiples are opposite relationships:

  • Factors: Numbers that divide into a given number evenly (e.g., factors of 12 are 1, 2, 3, 4, 6, 12)
  • Multipliers: Numbers you get by multiplying a given number by integers (e.g., multiples of 3 are 3, 6, 9, 12, 15...)

Think of it this way: 3 is a factor of 12 because 12 ÷ 3 = 4 (whole number). Meanwhile, 12 is a multiple of 3 because 3 × 4 = 12.

Every number has a finite set of factors but infinitely many multiples.

Question: How can I quickly tell if a number is prime? I'm having trouble identifying prime numbers.

Answer: A prime number has exactly two factors: 1 and itself. Here's a quick method:

  • Check if the number is less than 2 (not prime)
  • Check if it's 2 (prime - the only even prime)
  • Check if it's even (not prime, except 2)
  • Test divisibility by odd numbers up to √n

For example, to check if 17 is prime:

• It's not even

• √17 ≈ 4.1, so test 3: 17 ÷ 3 = 5.67 (not whole)

• Since no factors were found up to √17, 17 is prime!