Number Theory - Sixth Grade
Complete Notes & Formulas
1. Prime and Composite Numbers
Definitions
Prime Number: A whole number greater than 1 that has exactly 2 factors: 1 and itself
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Composite Number: A whole number greater than 1 that has more than 2 factors
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...
Special Case: 1 is NEITHER prime nor composite
Remember: 2 is the only EVEN prime number!
Prime Numbers 1-100
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
Total: 25 prime numbers between 1 and 100
How to Identify Prime or Composite
Example: Is 17 prime or composite?
Test: Try to divide 17 by prime numbers less than √17 ≈ 4.1
17 ÷ 2 = 8.5 (not a whole number)
17 ÷ 3 = 5.67... (not a whole number)
Only factors are 1 and 17
Answer: 17 is PRIME
2. Identify Factors & Factor Pairs
What are Factors?
Factors are whole numbers that divide evenly into another number (with no remainder).
If a × b = n, then a and b are factors of n
Example 1: Find All Factors of 24
Method: Find all pairs that multiply to 24
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factor Pairs
Factor pairs are two numbers that multiply together to give the original number.
Factor pairs of 24:
(1, 24), (2, 12), (3, 8), (4, 6)
3. Prime Factorization
What is Prime Factorization?
Prime factorization is writing a number as a product of prime numbers only.
Every composite number can be written as a unique product of prime numbers
Method 1: Factor Tree
Example: Prime factorization of 60
60
/ \
6 10
/ \ / \
2 3 2 5
Prime factors: 2, 2, 3, 5
60 = 2 × 2 × 3 × 5
Method 2: Division Method
Example: Prime factorization of 72
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
| 1
72 = 2 × 2 × 2 × 3 × 3
Prime Factorization with Exponents
Use exponents for repeated prime factors
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
4. Greatest Common Factor (GCF)
What is GCF?
The GREATEST COMMON FACTOR (GCF) is the largest number that divides evenly into two or more numbers
Method 1: Listing Factors
Example: Find GCF of 24 and 36
Step 1: List all factors
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Step 2: Identify common factors
Common factors: 1, 2, 3, 4, 6, 12
Step 3: Choose the greatest
GCF = 12
Method 2: Prime Factorization
Example: Find GCF of 24 and 36
Step 1: Prime factorization
24 = 2³ × 3
36 = 2² × 3²
Step 2: Take common primes with lowest exponent
Common: 2² and 3¹
Step 3: Multiply
GCF = 2² × 3 = 4 × 3 = 12
GCF = 12
GCF of Three or Four Numbers
Example: Find GCF of 12, 18, and 24
12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3
Common to all: 2¹ × 3¹
GCF = 2 × 3 = 6
5. Least Common Multiple (LCM)
What is LCM?
The LEAST COMMON MULTIPLE (LCM) is the smallest number that is a multiple of two or more numbers
Method 1: Listing Multiples
Example: Find LCM of 4 and 6
Step 1: List multiples
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Step 2: Find smallest common multiple
LCM = 12
Method 2: Prime Factorization
Example: Find LCM of 12 and 18
Step 1: Prime factorization
12 = 2² × 3
18 = 2 × 3²
Step 2: Take all primes with highest exponent
Highest power of 2: 2²
Highest power of 3: 3²
Step 3: Multiply
LCM = 2² × 3² = 4 × 9 = 36
LCM = 36
LCM of Three or Four Numbers
Example: Find LCM of 4, 6, and 8
4 = 2²
6 = 2 × 3
8 = 2³
Highest powers: 2³ × 3¹
LCM = 8 × 3 = 24
6. GCF and LCM Word Problems
When to Use GCF vs LCM
| Use GCF when: | Use LCM when: |
|---|---|
| Dividing into equal groups Finding the largest size Splitting/distributing equally Greatest number that works | Finding when events occur together Smallest number that works Patterns repeating Common timing |
Example 1: GCF Problem
Problem: Maria has 24 red flowers and 36 white flowers. She wants to make identical bouquets with the same number of red and white flowers in each. What is the greatest number of bouquets she can make?
Keyword: "greatest number" → Use GCF
Find GCF of 24 and 36
GCF(24, 36) = 12
Answer: 12 bouquets
Each bouquet will have 2 red flowers (24÷12) and 3 white flowers (36÷12)
Example 2: LCM Problem
Problem: Bus A arrives every 6 minutes and Bus B arrives every 8 minutes. If they both arrive at 9:00 AM, when will they next arrive at the same time?
Keyword: "same time" → Use LCM
Find LCM of 6 and 8
LCM(6, 8) = 24
Answer: 24 minutes later, at 9:24 AM
7. Sort Factors of Numerical Expressions
Understanding Factors in Expressions
A factor of an expression divides the expression evenly (no remainder).
Example: Which are factors of 36?
Given numbers: 2, 5, 6, 7, 9, 11
Test each:
36 ÷ 2 = 18 ✓ (Factor)
36 ÷ 5 = 7.2 ✗ (Not a factor)
36 ÷ 6 = 6 ✓ (Factor)
36 ÷ 7 = 5.14... ✗ (Not a factor)
36 ÷ 9 = 4 ✓ (Factor)
36 ÷ 11 = 3.27... ✗ (Not a factor)
Factors: 2, 6, 9
Not factors: 5, 7, 11
Quick Reference: Number Theory
| Term | Definition | Example |
|---|---|---|
| Prime | Exactly 2 factors (1 and itself) | 7 (factors: 1, 7) |
| Composite | More than 2 factors | 12 (factors: 1,2,3,4,6,12) |
| Factor | Divides evenly (no remainder) | 4 is a factor of 12 |
| GCF | Greatest common factor | GCF(12, 18) = 6 |
| LCM | Least common multiple | LCM(4, 6) = 12 |
💡 Important Tips to Remember
✓ 1 is neither prime nor composite
✓ 2 is the only even prime number
✓ Prime factorization is unique for every number
✓ Use exponents for repeated prime factors
✓ GCF: Take common primes with lowest exponent
✓ LCM: Take all primes with highest exponent
✓ GCF is always ≤ the smaller number
✓ LCM is always ≥ the larger number
✓ GCF × LCM = Product of the two numbers (for 2 numbers)
✓ All even numbers > 2 are composite
🧠 Memory Tricks & Strategies
Prime vs Composite:
PRIME = "Private" (only 1 and itself can divide it)
COMPOSITE = "Composed of" (made of multiple factors)
GCF vs LCM:
GCF = "Goes into" (Greatest that GOES INTO both)
LCM = "Lands on" (Least where both LAND ON)
Prime Factorization:
"Break it down until all are prime!"
Factor Pairs:
"Two by two, they multiply too!"
Remember "2":
2 is the ONLY even prime - all others are composite!
GCF for Word Problems:
"Greatest, Divide, Split, Groups" = GCF
LCM for Word Problems:
"Least, Together, Same time, Pattern" = LCM
Master Number Theory! 🔢 ✨ 📐
Understanding primes, factors, GCF, and LCM is the foundation of algebra!