Factors · 3 min read
Prime Factorization · Greatest Common Factor (GCF) · Least Common Multiple (LCM)
Prime factorization expresses a composite number as a product of prime factors. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).
Factor Tree Method
Start with the number and break it into any two factors. Continue breaking down composite factors until all factors are prime.
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Prime factorization: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Division Method
Repeatedly divide by the smallest prime that divides evenly.
- Find the prime factorization of 84.
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
Prime factorization: 84 = 2² × 3 × 7
Writing Prime Factorizations
Express prime factorizations using exponents for repeated factors: 72 = 2³ × 3² 100 = 2² × 5² 180 = 2² × 3² × 5
The greatest common factor of two or more numbers is the largest number that divides evenly into all of them.
Listing Method
List all factors of each number and identify the largest common factor.
GCF = 12
Prime Factorization Method
Find the prime factorization of each number. The GCF is the product of all common prime factors, each raised to the lowest power that appears.
- Find GCF(48, 180).
- 48 = 2⁴ × 3
- 180 = 2² × 3² × 5
Common primes: 2 and 3
Lowest powers: 2² and 3¹ GCF = 2² × 3 = 4 × 3 = 12
Applications of GCF
GCF is used in:
Simplifying fractions (divide numerator and denominator by GCF)
Distributing items into equal groups
Finding dimensions for cutting materials without waste
The least common multiple of two or more numbers is the smallest positive number that is a multiple of all of them.
Listing Method
List multiples of each number until you find the smallest common multiple.
Multiples of 6: 6, 12, 18, 24, 30, 36...
Multiples of 8: 8, 16, 24, 32, 40...
LCM = 24
Prime Factorization Method
Find the prime factorization of each number. The LCM is the product of all prime factors that appear, each raised to the highest power that appears.
- Find LCM(12, 18).
- 12 = 2² × 3
- 18 = 2 × 3²
All primes: 2, 3
Highest powers: 2², 3² LCM = 2² × 3² = 4 × 9 = 36
Relationship Between GCF and LCM
For any two positive integers a and b: GCF(a, b) × LCM(a, b) = a × b
This relationship provides an alternative method for finding LCM: LCM(a, b) = (a × b) ÷ GCF(a, b)
- Find LCM(15, 20).
- GCF(15, 20) = 5
- LCM = (15 × 20) ÷ 5 = 300 ÷ 5 = 60
Applications of LCM
LCM is used in:
Finding common denominators for adding fractions
Solving problems involving repeating cycles
Scheduling events that occur at different intervals
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