Factoring Numbers: A Complete Guide to the Fundamentals
Understanding how to factor numbers is one of the most fundamental skills in mathematics. Whether you're simplifying fractions, solving equations, or tackling advanced number theory, factoring is an essential tool. This comprehensive guide will take you from the basics of what factors are to advanced techniques for finding them efficiently.
What are Factors?
A factor (or divisor) of a number is a whole number that divides evenly into that number, leaving no remainder.
Definition
For integers a and b, we say "a is a factor of b" if:
b ÷ a = c (where c is a whole number with no remainder)
Equivalently: a × c = b
Example: Factors of 12
12 ÷ 1 = 12 ✓ So 1 and 12 are factors
12 ÷ 2 = 6 ✓ So 2 and 6 are factors
12 ÷ 3 = 4 ✓ So 3 and 4 are factors
12 ÷ 4 = 3 ✓ Already found
12 ÷ 5 = 2.4 ✗ Not a whole number
12 ÷ 6 = 2 ✓ Already found
Factors of 12: 1, 2, 3, 4, 6, 12
Key Properties of Factors
- Every number has at least two factors: 1 and itself
- Factors come in pairs: If a × b = n, then both a and b are factors
- Factors are always less than or equal to the number (except for negative factors)
- 1 is a factor of every number
- The number itself is always a factor
Prime vs. Composite Numbers
Understanding the difference between prime and composite numbers is crucial for factoring.
Prime Numbers
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.
Examples:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Key Facts:
- 2 is the only even prime number
- 2 is the smallest prime number
- There are infinitely many prime numbers
- Primes are the "building blocks" of all numbers
Composite Numbers
A composite number is a natural number greater than 1 that has more than two factors.
Examples:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25...
Key Facts:
- Can be factored into smaller whole numbers
- Every composite number can be expressed as a product of primes
- The smallest composite number is 4
Special Cases
The number 1:
- Neither prime nor composite
- Has only one factor: itself
- Called a "unit" in number theory
The number 0:
- Not considered in prime/composite classification
- Has infinitely many factors (every number divides 0)
Quick Recognition
How to identify primes quickly:
- Even numbers: Only 2 is prime; all others are composite
- Ends in 5: Only 5 is prime; all others divisible by 5
- Small primes to memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Finding All Factors of a Number
Method 1: Systematic Division (Trial Division)
Test each number from 1 up to the number itself.
Example: Find factors of 36
36 ÷ 1 = 36 ✓ Factors: 1, 36
36 ÷ 2 = 18 ✓ Factors: 2, 18
36 ÷ 3 = 12 ✓ Factors: 3, 12
36 ÷ 4 = 9 ✓ Factors: 4, 9
36 ÷ 5 = 7.2 ✗
36 ÷ 6 = 6 ✓ Factor: 6 (only once)
All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Method 2: Pairs Method (More Efficient)
Only test up to the square root of the number. Each factor found gives you a pair!
Why? If a × b = n and a ≤ √n, then b ≥ √n. So we find all pairs by only checking up to √n.
Example: Find factors of 48
√48 ≈ 6.93, so test up to 6
1 × 48 = 48 → Factors: 1, 48
2 × 24 = 48 → Factors: 2, 24
3 × 16 = 48 → Factors: 3, 16
4 × 12 = 48 → Factors: 4, 12
6 × 8 = 48 → Factors: 6, 8
All factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Perfect Squares Warning:
For perfect squares, don't list the square root twice!
36: 1×36, 2×18, 3×12, 4×9, 6×6
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
(List 6 only once, not twice)
Method 3: Using Prime Factorization
This advanced method will be covered in detail later.
Use our number factorization calculator to find all factors instantly.
Prime Factorization
Prime factorization is expressing a number as a product of prime numbers.
The Fundamental Theorem of Arithmetic
Every integer greater than 1 can be expressed uniquely as a product of prime numbers (ignoring order).
Example:
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
This is the only way to express 60 as a product of primes (except rearranging).
Method 1: Factor Tree
Visual method: keep breaking down factors until all are prime.
Example: Prime factorization of 60
60
/ \
2 30
/ \
2 15
/ \
3 5
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Example: Prime factorization of 144
144
/ \
2 72
/ \
2 36
/ \
2 18
/ \
2 9
/ \
3 3
144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3²
Method 2: Repeated Division
Divide by the smallest prime repeatedly until you reach 1.
Example: Prime factorization of 180
2 | 180
2 | 90
3 | 45
3 | 15
5 | 5
| 1
180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
Step-by-step:
180 ÷ 2 = 90 (2 is a prime factor)
90 ÷ 2 = 45 (2 again)
45 ÷ 3 = 15 (now divisible by 3)
15 ÷ 3 = 5 (3 again)
5 is prime (done!)
Method 3: Continuous Division Table
More organized version of repeated division.
Example: Prime factorization of 252
| Prime | Quotient |
|-------|----------|
| 2 | 252 |
| 2 | 126 |
| 3 | 63 |
| 3 | 21 |
| 7 | 7 |
| | 1 |
252 = 2² × 3² × 7
Writing Prime Factorization
Standard form: Use exponents for repeated factors
72 = 2³ × 3²
Expanded form: List all prime factors
72 = 2 × 2 × 2 × 3 × 3
Both are correct, but exponent form is more concise.
Try our prime factorization calculator for instant results.
Divisibility Rules
Quick tests to determine if a number is divisible by small primes without actually dividing.
Rule for 2
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8 (is even)
Examples:
- 124 → ends in 4 → divisible by 2 ✓
- 357 → ends in 7 → not divisible by 2 ✗
Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3
Examples:
- 123 → 1+2+3 = 6 → 6÷3 = 2 → divisible by 3 ✓
- 456 → 4+5+6 = 15 → 15÷3 = 5 → divisible by 3 ✓
- 128 → 1+2+8 = 11 → 11÷3 ≠ whole → not divisible by 3 ✗
Rule for 4
A number is divisible by 4 if the last two digits form a number divisible by 4
Examples:
- 1,236 → 36÷4 = 9 → divisible by 4 ✓
- 5,128 → 28÷4 = 7 → divisible by 4 ✓
- 1,234 → 34÷4 = 8.5 → not divisible by 4 ✗
Rule for 5
A number is divisible by 5 if it ends in 0 or 5
Examples:
- 125 → ends in 5 → divisible by 5 ✓
- 450 → ends in 0 → divisible by 5 ✓
- 123 → ends in 3 → not divisible by 5 ✗
Rule for 6
A number is divisible by 6 if it's divisible by both 2 AND 3
Examples:
- 18 → even (÷2) and 1+8=9 (÷3) → divisible by 6 ✓
- 24 → even (÷2) and 2+4=6 (÷3) → divisible by 6 ✓
- 15 → odd (✗2) → not divisible by 6 ✗
Rule for 7
Double the last digit and subtract from the rest. If result is divisible by 7, so is the original
Example: Is 343 divisible by 7?
34 - (3×2) = 34 - 6 = 28
28 ÷ 7 = 4 ✓
So 343 is divisible by 7
Rule for 8
A number is divisible by 8 if the last three digits form a number divisible by 8
Examples:
- 1,216 → 216÷8 = 27 → divisible by 8 ✓
- 5,000 → 000÷8 = 0 → divisible by 8 ✓
Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9
Examples:
- 729 → 7+2+9 = 18 → 18÷9 = 2 → divisible by 9 ✓
- 456 → 4+5+6 = 15 → 15÷9 ≠ whole → not divisible by 9 ✗
Rule for 10
A number is divisible by 10 if it ends in 0
Examples:
- 120 → ends in 0 → divisible by 10 ✓
- 125 → ends in 5 → not divisible by 10 ✗
Quick Reference Table
| Divisor | Rule | |---------|------| | 2 | Last digit is even | | 3 | Sum of digits ÷ 3 | | 4 | Last 2 digits ÷ 4 | | 5 | Ends in 0 or 5 | | 6 | Divisible by 2 AND 3 | | 7 | Special rule (double last, subtract) | | 8 | Last 3 digits ÷ 8 | | 9 | Sum of digits ÷ 9 | | 10 | Ends in 0 |
Factor Counting
How Many Factors?
If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ (prime factorization):
Number of factors = (a₁ + 1)(a₂ + 1)...(aₖ + 1)
Example: How many factors does 72 have?
72 = 2³ × 3²
Number of factors = (3+1)(2+1) = 4 × 3 = 12
Let's verify by listing:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Count: 12 ✓
Example: How many factors does 100 have?
100 = 2² × 5²
Number of factors = (2+1)(2+1) = 3 × 3 = 9
Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100
Count: 9 ✓
Sum of Factors
If n = p₁^a₁ × p₂^a₂ (for simplicity):
Sum of factors = [(p₁^(a₁+1) - 1)/(p₁-1)] × [(p₂^(a₂+1) - 1)/(p₂-1)]
Example: Sum of factors of 12
12 = 2² × 3
Sum = [(2³-1)/(2-1)] × [(3²-1)/(3-1)]
= [(8-1)/1] × [(9-1)/2]
= 7 × 4
= 28
Verify: 1+2+3+4+6+12 = 28 ✓
Applications of Number Factorization
1. Simplifying Fractions
Problem: Simplify 48/72
48 = 2⁴ × 3
72 = 2³ × 3²
GCF = 2³ × 3 = 24
48/72 = (48÷24)/(72÷24) = 2/3
2. Finding LCM and GCF
Problem: Find GCF and LCM of 60 and 84
60 = 2² × 3 × 5
84 = 2² × 3 × 7
GCF = 2² × 3 = 12
LCM = 2² × 3 × 5 × 7 = 420
3. Perfect Squares
A number is a perfect square if all exponents in its prime factorization are even.
Examples:
36 = 2² × 3² → all even → 36 = 6² ✓
50 = 2 × 5² → 2 has odd exponent → not perfect square ✗
144 = 2⁴ × 3² → all even → 144 = 12² ✓
4. Perfect Cubes
A number is a perfect cube if all exponents in its prime factorization are multiples of 3.
Examples:
27 = 3³ → exponent is 3 → 27 = 3³ ✓
64 = 2⁶ → exponent is 6 (multiple of 3) → 64 = 4³ ✓
72 = 2³ × 3² → 3² has exponent not divisible by 3 ✗
5. Solving Word Problems
Problem: A rectangular garden has area 180 square meters. What are possible whole-number dimensions?
180 = 2² × 3² × 5
Factor pairs of 180:
1 × 180, 2 × 90, 3 × 60, 4 × 45, 5 × 36,
6 × 30, 9 × 20, 10 × 18, 12 × 15
All are valid dimensions!
Practice Problems
Beginner:
- List all factors of 24
- Is 17 prime or composite?
- Find the prime factorization of 50
Intermediate:
- How many factors does 90 have?
- Is 4,356 divisible by 6?
- Find GCF(48, 60)
Advanced:
- What is the smallest number divisible by 2, 3, 4, 5, and 6?
- How many factors does 2⁴ × 3³ × 5² have?
- Find three consecutive composite numbers
Answers:
- 1, 2, 3, 4, 6, 8, 12, 24
- Prime (only factors are 1 and 17)
- 50 = 2 × 5²
- 90 = 2 × 3² × 5 → (1+1)(2+1)(1+1) = 12 factors
- Yes (even and 4+3+5+6=18, divisible by 3)
- 48 = 2⁴ × 3, 60 = 2² × 3 × 5, GCF = 2² × 3 = 12
- LCM(2,3,4,5,6) = 60
- (4+1)(3+1)(2+1) = 5 × 4 × 3 = 60 factors
- Example: 8, 9, 10 (or 14, 15, 16, or 24, 25, 26, etc.)
Tips for Mastery
Study Strategies
1. Memorize Small Primes: Know the first 20 primes by heart: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
2. Practice Divisibility Rules: Test numbers randomly throughout the day
3. Factor Factor Tree: Draw trees for numbers 1-100 to build intuition
4. Look for Patterns:
- Powers of 2: 2, 4, 8, 16, 32, 64, 128...
- Powers of 3: 3, 9, 27, 81, 243...
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Common Mistakes to Avoid
✗ Forgetting that 1 is a factor of every number ✗ Thinking 1 is prime (it's not!) ✗ Listing factors in random order (use ascending order) ✗ Missing factor pairs when using the pairs method ✗ Listing the square root twice for perfect squares
Using Technology
While learning, use our calculators to verify your work:
- Number Factorization Calculator - Find all factors
- Prime Factorization Calculator - Get prime factorization
- GCF Calculator - Find greatest common factors
Conclusion
Understanding number factorization is fundamental to mathematics. Key takeaways:
Factors:
- Numbers that divide evenly
- Come in pairs
- Always include 1 and the number itself
Prime vs. Composite:
- Primes have exactly 2 factors
- Composites have more than 2 factors
- 1 is neither prime nor composite
Prime Factorization:
- Unique for every number
- Uses only prime numbers
- Foundation for many mathematical operations
Practical Skills:
- Divisibility rules save time
- Factor trees visualize the process
- Prime factorization solves many problems
Applications:
- Simplifying fractions
- Finding GCF and LCM
- Solving equations
- Real-world problem solving
Mastering number factorization builds a foundation for algebra, number theory, and advanced mathematics. Practice regularly, use systematic methods, and leverage online tools to verify your work. With time and effort, factoring numbers will become second nature, opening doors to more advanced mathematical concepts and problem-solving techniques.
Remember: every complex number is just a product of simple primes. Understanding this fundamental truth empowers you to break down and conquer mathematical challenges at any level!