Identifying Factors Worksheet PDF | 4th Grade Math Exercises

Master factors, multiples, and patterns: learn to identify factors, understand divisibility rules, and solve problems with step-by-step solutions.

Solution: Exercises 1 to 3
1 Finding Factors
Exercise 1
Find all factors of 24. List them in order from smallest to largest.
Definition:

Factor: A number that divides another number evenly without leaving a remainder.

Factor pairs: Two numbers that multiply together to make the target number.

Method to find factors:
  1. Start with 1 and check if it divides the number evenly
  2. Continue checking each number up to the square root of the target
  3. For each divisor found, record its pair (the quotient)
  4. List all unique factors found
Target Number
24
Factor Pairs
1×24, 2×12, 3×8, 4×6
Step 1: Start with 1

24 ÷ 1 = 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 found 6 as a factor (from 4×6), so we stop here

1
2
3
4
6
8
12
24
1 × 24 = 24
→
2 × 12 = 24
→
3 × 8 = 24
→
4 × 6 = 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:

• Division Rule: A number is a factor if division results in a whole number

• Pair Rule: Factors come in pairs that multiply to the target number

• Square Root Rule: Only check up to the square root to find all factors efficiently

Tip: Every number has at least two factors: 1 and itself.
Tip: Factors are always less than or equal to the number.
2 Factor Identification
Exercise 2
Which of these numbers are factors of 36? Circle them: 2, 3, 4, 5, 6, 7, 8, 9
Definition:

Factor Test: To check if a number is a factor, divide the target number by the potential factor. If the result is a whole number, it's a factor.

Target Number
36
Check Numbers
2, 3, 4, 5, 6, 7, 8, 9
Step 1: Test 2

36 ÷ 2 = 18 (whole number) → 2 is a factor

Step 2: Test 3

36 ÷ 3 = 12 (whole number) → 3 is a factor

Step 3: Test 4

36 ÷ 4 = 9 (whole number) → 4 is a factor

Step 4: Test 5

36 ÷ 5 = 7.2 (not whole) → 5 is not a factor

Step 5: Test 6

36 ÷ 6 = 6 (whole number) → 6 is a factor

Step 6: Test 7

36 ÷ 7 = 5.14... (not whole) → 7 is not a factor

Step 7: Test 8

36 ÷ 8 = 4.5 (not whole) → 8 is not a factor

Step 8: Test 9

36 ÷ 9 = 4 (whole number) → 9 is a factor

✓ 2
✓ 3
✓ 4
✗ 5
✓ 6
✗ 7
✗ 8
✓ 9
2 × 18 = 36
3 × 12 = 36
4 × 9 = 36
6 × 6 = 36
Factors of 36 from the list: 2, 3, 4, 6, 9
Final answer:

The factors of 36 from the given list are: 2, 3, 4, 6, 9

Applied rules:

• Division Test: A number is a factor if division results in no remainder

• Factor Verification: Multiply the factor by its pair to confirm

• Systematic Testing: Test each number individually

3 Factor Pairs
Exercise 3
Write all factor pairs for the number 18.
Definition:

Factor Pair: Two numbers that multiply together to equal the target number.

Target Number
18
Factor Pairs
Find all pairs
Step 1: Find all factors of 18

18 ÷ 1 = 18 → 1 and 18

18 ÷ 2 = 9 → 2 and 9

18 ÷ 3 = 6 → 3 and 6

18 ÷ 4 = 4.5 → not a factor

18 ÷ 5 = 3.6 → not a factor

18 ÷ 6 = 3 → already found (3×6)

Step 2: List the factor pairs

(1, 18), (2, 9), (3, 6)

Step 3: Verify each pair

1 × 18 = 18 ✓

2 × 9 = 18 ✓

3 × 6 = 18 ✓

(1, 18)
(2, 9)
(3, 6)
1 × 18 = 18
2 × 9 = 18
3 × 6 = 18
Factor pairs of 18: (1, 18), (2, 9), (3, 6)
Final answer:

The factor pairs of 18 are: (1, 18), (2, 9), (3, 6)

Applied rules:

• Factor Pair Rule: Each factor has a corresponding pair

• Verification Rule: Always multiply to confirm the product

• Order Rule: List pairs in ascending order of the first number

Rules and methods, laws,...
If a × b = n, then a and b are factors of n
Factor Definition
Divisibility by 2
Even numbers
Last digit is 0, 2, 4, 6, or 8
Divisibility by 3
Sum of digits
Sum of digits is divisible by 3
Divisibility by 5
Ends in 0 or 5
Last digit is 0 or 5
Key definitions:

Factor: A number that divides another number evenly

Multiple: A number that can be divided by another number evenly

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

Composite Number: A number with more than two factors

Complete methodology:
  1. Understand the concept: A factor divides the target number evenly
  2. Systematic approach: Start with 1 and work up to the square root
  3. Pair method: For each factor found, identify its pair
  4. Verification: Multiply factor pairs to confirm
Tip 1: The smallest factor is always 1, the largest is the number itself.
Tip 2: Factors come in pairs that multiply to the original number.
Tip 3: Only check up to the square root to find all factors efficiently.
Tip 4: If a number has only 2 factors, it's prime.
Common errors: Forgetting 1 and the number itself as factors, missing factor pairs, miscalculating division.
Key insight: The number of factors is finite and can be listed completely.
Divisibility rules to know:

• By 2: Last digit is even (0, 2, 4, 6, 8)

• By 3: Sum of digits is divisible by 3

• By 4: Last two digits form a number divisible by 4

• By 5: Last digit is 0 or 5

• By 6: Divisible by both 2 and 3

• By 9: Sum of digits is divisible by 9

• By 10: Last digit is 0

Solution: Exercises 4 to 5
4 True/False Factors
Exercise 4
Determine if each statement is True or False:
a) 5 is a factor of 20
b) 6 is a factor of 25
c) 8 is a factor of 32
d) 7 is a factor of 42
Definition:

Factor Test: A number is a factor if division results in a whole number with no remainder.

Statement A
20 ÷ 5 = 4
Result A
True
Step 1: Test statement a) 5 is a factor of 20

20 ÷ 5 = 4 (whole number) → True

Step 2: Test statement b) 6 is a factor of 25

25 ÷ 6 = 4.166... (not whole) → False

Step 3: Test statement c) 8 is a factor of 32

32 ÷ 8 = 4 (whole number) → True

Step 4: Test statement d) 7 is a factor of 42

42 ÷ 7 = 6 (whole number) → True

a) True
b) False
c) True
d) True
5 × 4 = 20 ✓
6 × ? ≠ 25 ✗
8 × 4 = 32 ✓
7 × 6 = 42 ✓
a) True, b) False, c) True, d) True
Final answer:

a) True, b) False, c) True, d) True

Applied rules:

• Division Test: Check if division results in a whole number

• Verification: Multiply to confirm the relationship

• Remainder Check: Factors produce zero remainder

5 Multiple Factors
Exercise 5
List all factors of 48 and identify which ones are also factors of 24.
Definition:

Common Factors: Numbers that are factors of two or more different numbers.

Number 1
48
Number 2
24
Common
Shared factors
Step 1: Find factors of 48

48 ÷ 1 = 48 → 1, 48

48 ÷ 2 = 24 → 2, 24

48 ÷ 3 = 16 → 3, 16

48 ÷ 4 = 12 → 4, 12

48 ÷ 6 = 8 → 6, 8

48 ÷ 7 = 6.85... → not a factor

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Find factors of 24

From Exercise 1: 1, 2, 3, 4, 6, 8, 12, 24

Step 3: Identify common factors

Compare the two lists: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 48
1
2
3
4
6
8
12
16
24
48
Common with 24
1
2
3
4
6
8
12
24
Common factors of 48 and 24: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Common factors with 24: 1, 2, 3, 4, 6, 8, 12, 24

Applied rules:

• Comparison Method: Find factors of each number separately, then compare

• Commonality: Shared factors appear in both lists

• Subset Relationship: If one number is a factor of another, all its factors are also factors of the larger number

Factors and Multiples Comprehensive Guide
If a × b = n, then a and b are factors of n
Factor Definition
Key definitions:

Factor: A number that divides another number evenly without remainder

Multiple: The product of a number and any integer

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

Composite Number: A number with more than two factors

Prime Factorization: Expressing a number as a product of prime factors

Complete methodology:
  1. Systematic Approach: Start with 1 and check each number up to √n
  2. Factor Pairs: For each divisor d, n÷d gives the paired factor
  3. Verification: Multiply factor pairs to confirm they equal n
  4. Organization: List factors in ascending order
Tip 1: Use divisibility rules to quickly identify potential factors.
Tip 2: Stop checking at the square root - factors come in pairs.
Tip 3: The greatest common factor (GCF) is the largest factor shared by multiple numbers.
Tip 4: The least common multiple (LCM) is the smallest multiple shared by multiple numbers.
Important Properties: Every number has at least two factors (1 and itself), factors are always ≤ the number, and they come in pairs.
Patterns: Even numbers always have 2 as a factor, multiples of 10 end in 0, etc.
Divisibility rules:

• By 2: Last digit is 0, 2, 4, 6, or 8

• By 3: Sum of digits is divisible by 3

• By 4: Last two digits form a number divisible by 4

• By 5: Last digit is 0 or 5

• By 6: Divisible by both 2 and 3

• By 9: Sum of digits is divisible by 9

• By 10: Last digit is 0

Questions & Answers

Question: I'm confused about how to find all the factors of a number. It seems like there are so many possibilities. How do I make sure I don't miss any?

Answer: Great question! There's a systematic way to find all factors that ensures you don't miss any:

  1. Start small: Begin with 1 and see if it divides your number evenly
  2. Work up systematically: Check 2, 3, 4, and so on
  3. Stop at the square root: Once you reach the square root of your number, you've found all unique factors
  4. Record pairs: When you find a factor, write down both it and its pair

Example for 20:

  • 1 divides 20 → factors: 1, 20
  • 2 divides 20 → factors: 2, 10
  • 3 doesn't divide 20
  • 4 divides 20 → factors: 4, 5
  • 5 is already found, so stop

This gives us: 1, 2, 4, 5, 10, 20. All factors found!

The key is being systematic and stopping at the square root (√20 ≈ 4.47).

Question: What's the difference between factors and multiples? They seem similar but I know they're different.

Answer: Factors and multiples are related but opposite concepts:

Factors:

  • Numbers that divide into another number evenly
  • Smaller than or equal to the original number
  • Finite number of factors for any number
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12

Multiples:

  • Numbers you get when you multiply the original number
  • Larger than or equal to the original number
  • Infinite number of multiples
  • Example: Multiples of 12 are 12, 24, 36, 48, 60, ...

Think of it this way: factors go INTO the number, multiples come FROM the number.

For number 6: Factors (go in) → 1, 2, 3, 6; Multiples (come from) → 6, 12, 18, 24, 30, ...

Question: My child struggles with recognizing patterns in factors. Are there any tricks or shortcuts to help them identify factors more quickly?

Answer: Yes, there are several helpful strategies and divisibility rules that make factor identification much easier:

Divisibility Rules (Shortcuts):

  • By 2: Number ends in 0, 2, 4, 6, or 8
  • By 3: Sum of digits is divisible by 3
  • By 5: Number ends in 0 or 5
  • By 9: Sum of digits is divisible by 9
  • By 10: Number ends in 0

Pattern Recognition Tips:

  • Every number has 1 and itself as factors
  • Factors come in pairs that multiply to the original number
  • Prime numbers have exactly two factors (1 and themselves)
  • Perfect squares have an odd number of factors

Example: For 36, knowing it ends in 6 means it's even (factor of 2), and 3+6=9 means it's divisible by 3 and 9.

These rules help narrow down possibilities quickly!