GCF, LCM, and Prime Factorization
Grade 6 · Math · Study Guide
Learn to find the greatest common factor, least common multiple, and prime factorization of numbers.
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. A composite number has more than two factors. Prime factorization breaks a number into a product of prime numbers. For 36: 36 = 4 × 9 = 2 × 2 × 3 × 3 = 2² × 3². Use a factor tree to find it systematically.
The Greatest Common Factor (GCF) of two numbers is the largest factor they share. To find it, list all factors of each number and find the biggest common one. Or use prime factorization — find all primes both numbers share and multiply them. GCF(12, 18): factors of 12 are 1,2,3,4,6,12 and factors of 18 are 1,2,3,6,9,18. The GCF is 6.
The Least Common Multiple (LCM) is the smallest number that both numbers divide into evenly. To find it, list multiples of each number until you find a match, or use prime factorization — take the highest power of every prime that appears in either number and multiply them. LCM(4, 6): multiples of 4 are 4,8,12... and multiples of 6 are 6,12... The LCM is 12.
GCF and LCM aren't just abstract — GCF helps you simplify fractions to lowest terms, and LCM helps you find common denominators when adding fractions. These are some of the most useful tools in all of math.
| Concept | What It Finds | Use It For | Example (12 and 18) |
|---|---|---|---|
| GCF | Largest shared factor | Simplifying fractions | GCF = 6 |
| LCM | Smallest shared multiple | Common denominators | LCM = 36 |
Key terms
- Prime Number
- A whole number greater than 1 with exactly two factors: 1 and itself
- Prime Factorization
- Writing a number as a product of its prime factors, like 12 = 2² × 3
- GCF (Greatest Common Factor)
- The largest factor shared by two or more numbers
- LCM (Least Common Multiple)
- The smallest positive number that is a multiple of two or more numbers
Flashcards
- Prime number
- Exactly two factors: 1 and itself — examples: 2, 3, 5, 7, 11, 13
- Composite number
- More than two factors — examples: 4, 6, 9, 12
- Prime factorization
- Breaking a number into only prime factors — e.g. 30 = 2 × 3 × 5
- Factor tree
- A diagram showing repeated factoring until all branches are prime
- GCF (Greatest Common Factor)
- Largest number that divides evenly into two or more numbers
- LCM (Least Common Multiple)
- Smallest number that two or more numbers divide into evenly
Practice questions
What is the prime factorization of 24?
- 2³ × 3
- 2² × 6
- 4 × 6
- 2 × 12
Show answer
Answer: 2³ × 3
24 = 2×12 = 2×2×6 = 2×2×2×3 = 2³×3. All factors are prime.
What is the GCF of 20 and 30?
- 10
- 5
- 60
- 2
Show answer
Answer: 10
Factors of 20: 1,2,4,5,10,20. Factors of 30: 1,2,3,5,6,10,15,30. Largest common factor is 10.
The number 1 is a prime number.
Show answer
Answer: False
Prime numbers must have EXACTLY two factors. 1 only has one factor (itself), so it is NOT prime.
The LCM of 4 and 6 is ___.
Show answer
Answer: 12
Multiples of 4: 4,8,12. Multiples of 6: 6,12. First common multiple is 12.
What is the GCF of 36 and 48?
- 12
- 6
- 144
- 3
Show answer
Answer: 12
36 = 2²×3², 48 = 2⁴×3. GCF uses minimum exponents: 2²×3 = 4×3 = 12.
Fun facts
- Every even number greater than 2 can be written as the sum of two prime numbers — this is called Goldbach's Conjecture, and no one has proven it yet for ALL even numbers!
- The number 1 is NOT prime because prime numbers must have exactly two factors, and 1 only has one factor: itself.
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