GCF and LCM Calculator
Common GCF & LCM Values
| Numbers | GCF | LCM |
|---|---|---|
| 4, 6 | 2 | 12 |
| 8, 12 | 4 | 24 |
| 15, 20 | 5 | 60 |
| 12, 18, 24 | 6 | 72 |
| 100, 150 | 50 | 300 |
Quick Reference
Related Calculators
Understanding GCF and LCM
What is GCF (Greatest Common Factor)?
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly.
Example: GCF of 24 and 36
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
GCF = 12
What is LCM (Least Common Multiple)?
The Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into evenly.
Example: LCM of 4 and 6
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36...
LCM = 12
Methods for Finding GCF
Prime Factorization Method
1. Find prime factorization of each number
2. Identify common prime factors
3. Multiply common factors (using lowest powers)
Euclidean Algorithm
1. Divide larger number by smaller
2. Replace larger with remainder
3. Repeat until remainder is 0
4. GCF is the last non-zero remainder
The GCF-LCM Relationship
There's an elegant mathematical relationship between GCF and LCM for any two numbers a and b:
This formula allows you to find LCM if you know GCF: LCM = (a × b) / GCF
Practical Applications
GCF Applications
- Simplifying fractions (divide both numerator and denominator by GCF)
- Distributing items into groups of equal size
- Finding the largest square tile that fits evenly in a room
- Reducing ratios to lowest terms
LCM Applications
- Adding and subtracting fractions (finding common denominators)
- Scheduling recurring events (finding when they coincide)
- Gear ratios and rotational mechanics
- Music theory (finding when rhythms align)
Example: Scheduling Problem
Real-World Problem
Bus A comes every 12 minutes, Bus B comes every 18 minutes. If both buses arrive at 8:00 AM, when will they next arrive together?
GCF and LCM for Multiple Numbers
To find the GCF or LCM of more than two numbers, apply the operation repeatedly. For example, GCF(a, b, c) = GCF(GCF(a, b), c).
| Numbers | GCF | LCM | GCF × LCM |
|---|---|---|---|
| 4, 6 | 2 | 12 | 24 = 4 × 6 |
| 12, 18 | 6 | 36 | 216 = 12 × 18 |
| 15, 25 | 5 | 75 | 375 = 15 × 25 |
| 8, 12, 20 | 4 | 120 | - |
Pro Tip: Coprime Numbers
Two numbers are coprime (or relatively prime) if their GCF is 1. For coprime numbers a and b: LCM(a, b) = a × b. Examples of coprime pairs: (8, 15), (7, 11), (9, 14).
Frequently Asked Questions
What is the difference between GCF and LCM?
GCF (Greatest Common Factor) is the largest number that divides all given numbers evenly. LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly. For example, for 12 and 18: GCF = 6 (largest shared factor), LCM = 36 (smallest shared multiple).
How do you find GCF using prime factorization?
Break each number into prime factors, then multiply the common prime factors using the lowest power of each. For 24 (2^3 x 3) and 36 (2^2 x 3^2): Common factors are 2 and 3. Use lowest powers: 2^2 x 3^1 = 12. So GCF(24,36) = 12.
When would I use LCM in real life?
LCM is useful for finding when recurring events coincide. For example: if Bus A comes every 12 minutes and Bus B every 18 minutes, LCM(12,18) = 36 tells you they'll arrive together every 36 minutes. It's also essential for adding fractions with different denominators.
Is there a shortcut formula relating GCF and LCM?
Yes! For any two numbers a and b: GCF(a,b) x LCM(a,b) = a x b. This means if you know the GCF, you can find LCM by: LCM = (a x b) / GCF. For example, with 12 and 18: GCF = 6, so LCM = (12 x 18) / 6 = 36.