Complete Guide: Finding Factors of Numbers in Grade 4

Master finding factors of numbers: comprehensive guide with detailed exercises, definitions, methods, and practice problems.

Complete Summary: Finding Factors
Comprehensive Summary
Key Definitions:

Factor: A number that divides another number evenly without remainder. If a × b = n, then both a and b are factors of n.

Multiple: A number that results from multiplying a factor by an integer. If a × b = n, then n is a multiple of both a and b.

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.

Composite Number: A number greater than 1 that has more than two factors.

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

Core Rules and Laws:

• Every number is divisible by 1 and itself

• Factors come in pairs: if a is a factor of n, then n/a is also a factor

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

• You only need to check up to √n to find all factors

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

Step-by-Step Method:
  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
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.
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.
Simple Example: Factors of 8

• 8 ÷ 1 = 8 → (1, 8)

• 8 ÷ 2 = 4 → (2, 4)

• 8 ÷ 3 = 2.67 (not whole)

• √8 ≈ 2.8, so stop here

• Factors: 1, 2, 4, 8

Advanced Example: Factors of 36

• 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

• Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Key Takeaways:

• Factors are numbers that divide evenly into another number

• Use systematic approach to find all factors efficiently

• Remember the pairing principle and stopping rule

• Practice with various numbers to build fluency

Examples and Exercises
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
Practice Problems
Problem 1
Find all factors of 15.
Problem 2
Find all factors of 28.
Problem 3
Find all factors of 40.
Problem 4
Find all factors of 35.
Detailed Solutions
Solution 1: Factors of 15
Step 1: Start with 1 and 15

15 ÷ 1 = 15 → 1 and 15 are factors

Step 2: Test 2

15 ÷ 2 = 7.5 (not whole)

Step 3: Test 3

15 ÷ 3 = 5 → 3 and 5 are factors

Step 4: Test 4

15 ÷ 4 = 3.75 (not whole)

Step 5: Stop at √15 ≈ 3.9

We already found 3 and 5, so we're done

Factors of 15: 1, 3, 5, 15
Solution 2: Factors of 28
Step 1: Start with 1 and 28

28 ÷ 1 = 28 → 1 and 28 are factors

Step 2: Test 2

28 ÷ 2 = 14 → 2 and 14 are factors

Step 3: Test 3

28 ÷ 3 = 9.33 (not whole)

Step 4: Test 4

28 ÷ 4 = 7 → 4 and 7 are factors

Step 5: Stop at √28 ≈ 5.3

We already found 4 and 7, so we're done

Factors of 28: 1, 2, 4, 7, 14, 28
Solution 3: Factors of 40
Step 1: Start with 1 and 40

40 ÷ 1 = 40 → 1 and 40 are factors

Step 2: Test 2

40 ÷ 2 = 20 → 2 and 20 are factors

Step 3: Test 3

40 ÷ 3 = 13.33 (not whole)

Step 4: Test 4

40 ÷ 4 = 10 → 4 and 10 are factors

Step 5: Test 5

40 ÷ 5 = 8 → 5 and 8 are factors

Step 6: Stop at √40 ≈ 6.3

We already found 5 and 8, so we're done

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Solution 4: Factors of 35
Step 1: Start with 1 and 35

35 ÷ 1 = 35 → 1 and 35 are factors

Step 2: Test 2

35 ÷ 2 = 17.5 (not whole)

Step 3: Test 3

35 ÷ 3 = 11.67 (not whole)

Step 4: Test 4

35 ÷ 4 = 8.75 (not whole)

Step 5: Test 5

35 ÷ 5 = 7 → 5 and 7 are factors

Step 6: Stop at √35 ≈ 5.9

We already found 5 and 7, so we're done

Factors of 35: 1, 5, 7, 35
Worksheet Practice Section
PDF Worksheet Structure
Organized Practice
Worksheet Components:

Instructions: Clear directions for completing the worksheet

Guided Examples: Step-by-step worked problems

Practice Problems: Exercises for independent work

Answer Key: Solutions for self-checking

Review Questions: Mixed problems for reinforcement

Worksheet Completion Strategy:
  1. Read instructions carefully before starting
  2. Review the guided examples to understand the format
  3. Work systematically through each problem
  4. Check your work using the answer key
  5. Practice regularly to build fluency
Tip 1: Use a systematic approach: start with 1, test 2, 3, 4... up to √n.
Tip 2: Write factors in pairs to avoid missing any.
Tip 3: Always verify by multiplying factor pairs to get the original number.
Tip 4: Circle or highlight your final answer for clarity.
Common Mistakes: Forgetting 1 and the number itself, missing factor pairs, going beyond √n unnecessarily.
Success Strategies: Practice with numbers 1-50, memorize divisibility rules, check your work systematically.
PDF Worksheet Format:

• Title and header information

• Clear problem statements

• Space for working out solutions

• Answer boxes for final responses

• Progress tracking sections

Visualization: Factor Patterns
Comparing Numbers and Their 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!