GCF and LCM Calculator
Calculate both Greatest Common Factor (GCD) and Least Common Multiple (LCM) simultaneously
Calculate GCF and LCM
Example Calculation
Find GCF and LCM of 12 and 18
Method: Euclidean Algorithm + Formula
Step 1 - Find GCF:
18 ÷ 12 = 1 remainder 6 → GCF(18, 12) = GCF(12, 6)
12 ÷ 6 = 2 remainder 0 → GCF(12, 6) = 6
Step 2 - Find LCM:
LCM(12, 18) = (12 × 18) ÷ GCF(12, 18) = 216 ÷ 6 = 36
Answer
GCF(12, 18) = 6
LCM(12, 18) = 36
Verification: 6 × 36 = 216 = 12 × 18 ✓
Key Relationships
Fundamental Formula
GCF(a,b) × LCM(a,b) = a × b
LCM Formula
LCM(a,b) = |a × b| ÷ GCF(a,b)
Coprime Numbers
If GCF(a,b) = 1, then LCM(a,b) = a × b
Calculation Methods
Euclidean Algorithm
Most efficient for large numbers
Prime Factorization
Shows mathematical structure clearly
Formula Method
Direct calculation using GCF×LCM formula
Understanding GCF and LCM
Greatest Common Factor (GCF)
The Greatest Common Factor, also known as Greatest Common Divisor (GCD), is the largest positive integer that divides all given numbers without leaving a remainder. It represents the highest common factor shared by the numbers.
Applications of GCF
- •Simplifying fractions to lowest terms
- •Distributing items equally into groups
- •Finding common denominators
- •Solving Diophantine equations
Least Common Multiple (LCM)
The Least Common Multiple is the smallest positive integer that is divisible by all given numbers. It represents the smallest number that all input numbers can divide evenly into.
Applications of LCM
- •Adding or subtracting fractions
- •Scheduling recurring events
- •Finding common time intervals
- •Solving word problems involving cycles
Important Properties
GCF Properties
GCF(a, a) = a
GCF(a, 1) = 1
GCF(a, 0) = a
GCF(a, b) = GCF(b, a)
LCM Properties
LCM(a, a) = a
LCM(a, 1) = a
LCM(a, b) = LCM(b, a)
LCM(a, b) ≥ max(a, b)
Real-World Examples
GCF Example: Package Distribution
You have 24 apples and 36 oranges to distribute equally into gift bags. What's the maximum number of identical bags you can make?
Answer: GCF(24, 36) = 12 bags (each with 2 apples and 3 oranges)
LCM Example: Meeting Schedule
Team A meets every 4 days, Team B every 6 days. If both teams meet today, when will they meet together again?
Answer: LCM(4, 6) = 12 days from today