Number Theory - Seventh Grade
Prime Factorization, GCF & LCM
1. Prime Factorization
What is Prime Factorization?
Prime factorization is expressing a number as a
PRODUCT OF PRIME NUMBERS
• Every composite number has a unique prime factorization
• Prime numbers cannot be factorized (they're already prime!)
Key Terms
Prime Number: A number with exactly TWO factors (1 and itself)
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...
Composite Number: A number with MORE than two factors
Examples: 4, 6, 8, 9, 10, 12, 14, 15...
Method 1: Factor Tree Method
Step 1: Write the number at the top
Step 2: Split into ANY two factors
Step 3: Continue factoring until ALL factors are prime
Step 4: Circle all prime numbers
Step 5: Multiply circled primes together
Example: Prime Factorization of 60
Factor Tree:
60
/ \
6 10
/ \ / \
2 3 2 5
All circled: 2, 3, 2, 5Prime Factorization: 60 = 2 × 2 × 3 × 5
Exponential Form: 60 = 2² × 3 × 5
Answer: 2² × 3 × 5
Method 2: Division Method
Divide by the smallest prime repeatedly
until the quotient becomes 1
Example: Prime Factorization of 72
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
1Prime Factorization: 72 = 2 × 2 × 2 × 3 × 3
Exponential Form: 72 = 2³ × 3²
Answer: 2³ × 3²
2. Greatest Common Factor (GCF)
What is GCF?
GCF is the LARGEST number that divides
evenly into two or more numbers
• Also called HCF (Highest Common Factor)
• GCF is always ≤ the smallest number
Method 1: Listing Factors
Step 1: List ALL factors of each number
Step 2: Identify COMMON factors
Step 3: Select the GREATEST common factor
Example: Find GCF of 30 and 42
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Common factors: 1, 2, 3, 6
Greatest common factor: 6
Answer: GCF(30, 42) = 6
Method 2: Prime Factorization
Step 1: Find prime factorization of each number
Step 2: Identify COMMON prime factors
Step 3: Use LOWEST power of each common prime
Step 4: Multiply common primes together
Example: Find GCF of 60 and 90
Prime Factorizations:
60 = 2² × 3 × 5
90 = 2 × 3² × 5
Common prime factors:
2 (lowest power: 2¹)
3 (lowest power: 3¹)
5 (lowest power: 5¹)
GCF: 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30
Answer: GCF(60, 90) = 30
Quick Tip: If two numbers have NO common factors other than 1, they are called "relatively prime" and their GCF = 1
3. Least Common Multiple (LCM)
What is LCM?
LCM is the SMALLEST number that is a
multiple of two or more numbers
• LCM is always ≥ the largest number
• Used for adding/subtracting fractions
Method 1: Listing Multiples
Step 1: List multiples of each number
Step 2: Identify COMMON multiples
Step 3: Select the SMALLEST common multiple
Example: Find LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20, 24, 28...
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36...
Least common multiple: 12
Answer: LCM(4, 6) = 12
Method 2: Prime Factorization
Step 1: Find prime factorization of each number
Step 2: List ALL prime factors that appear
Step 3: Use HIGHEST power of each prime
Step 4: Multiply all primes together
Example: Find LCM of 12 and 18
Prime Factorizations:
12 = 2² × 3
18 = 2 × 3²
All prime factors with highest powers:
2 (highest power: 2²)
3 (highest power: 3²)
LCM: 2² × 3² = 4 × 9 = 36
Answer: LCM(12, 18) = 36
LCM Formula (Using GCF)
LCM(a, b) = (a × b) ÷ GCF(a, b)
4. GCF and LCM Relationship
Important Formula
GCF(a, b) × LCM(a, b) = a × b
Example: Verify with 12 and 18
GCF(12, 18) = 6
LCM(12, 18) = 36
Check:
GCF × LCM = 6 × 36 = 216
a × b = 12 × 18 = 216 ✓
Formula verified!
Quick Comparison
| Aspect | GCF | LCM |
|---|---|---|
| Definition | Greatest Common Factor | Least Common Multiple |
| Type | Factor (divides) | Multiple (contains) |
| Size | ≤ smallest number | ≥ largest number |
| Prime Method | Lowest powers | Highest powers |
5. GCF and LCM Word Problems
When to Use GCF vs LCM
Use GCF when:
• Dividing/splitting things into EQUAL GROUPS
• Finding largest size that fits evenly
• Keywords: "largest," "greatest," "split," "arrange"
Use LCM when:
• Events happening at DIFFERENT INTERVALS
• Finding when things happen TOGETHER again
• Keywords: "least," "smallest," "both," "together," "at the same time"
GCF Word Problem Example
Problem: You have 24 apples and 36 oranges. What is the greatest number of fruit baskets you can make if each basket has the same number of apples and the same number of oranges?
Solution: Find GCF(24, 36)
24 = 2³ × 3
36 = 2² × 3²
GCF = 2² × 3 = 12
Check: 24 ÷ 12 = 2 apples per basket
36 ÷ 12 = 3 oranges per basket
Answer: 12 baskets
LCM Word Problem Example
Problem: Bus A arrives every 12 minutes. Bus B arrives every 18 minutes. If both buses arrive at 8:00 AM, when will they both arrive together again?
Solution: Find LCM(12, 18)
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36 minutes
8:00 AM + 36 minutes = 8:36 AM
Answer: 8:36 AM
Quick Reference: Number Theory Formulas
| Concept | Key Rule/Formula |
|---|---|
| Prime Factorization | Express as product of primes |
| GCF (Prime Method) | Multiply common primes with LOWEST powers |
| LCM (Prime Method) | Multiply all primes with HIGHEST powers |
| LCM Formula | LCM(a,b) = (a × b) ÷ GCF(a,b) |
| Relationship | GCF × LCM = a × b |
💡 Important Tips to Remember
✓ Prime factorization is unique - every number has only ONE
✓ Factor tree: Can start with any two factors
✓ GCF: Greatest Common Factor (divides evenly)
✓ LCM: Least Common Multiple (divisible by both)
✓ GCF ≤ smallest number, LCM ≥ largest number
✓ For GCF: Use LOWEST powers of common primes
✓ For LCM: Use HIGHEST powers of all primes
✓ GCF for splitting/dividing, LCM for cycles/timing
✓ Check your work: GCF × LCM = Product of numbers
✓ Use formula: If you know GCF, find LCM easily!
🧠 Memory Tricks & Strategies
Prime Factorization:
"Break it down to prime, factor tree every time!"
GCF:
"GCF is greatest, it divides the best - common and low powers pass the test!"
LCM:
"LCM is least, but bigger than both - high powers of primes, that's the oath!"
GCF vs LCM (Prime Method):
"GCF goes LOW, LCM goes HIGH - that's the rule, don't ask why!"
Word Problems:
"Split and divide? GCF's your guide! Together again? LCM's your friend!"
Relationship Formula:
"GCF times LCM equals a times b - this relationship is the key!"
Master Number Theory! 🔢 ➗ ✖️
Remember: Prime factorization is the key to GCF and LCM!