Common Factor Calculator

Common factors are numbers that divide evenly into two or more numbers. The Greatest Common Factor (GCF) is the largest of these common factors, while the Least Common Multiple (LCM) is the smallest number that is a multiple of all the given numbers.

Greatest Common Factor (GCF)

The GCF, also called Greatest Common Divisor (GCD), is the largest number that divides evenly into all given numbers. For example:

GCF of 12 and 18 is 6 (factors: 12 = 1,2,3,4,6,12 | 18 = 1,2,3,6,9,18)
GCF of 24, 36, and 48 is 12
GCF of 17 and 19 is 1 (they're both prime)

Least Common Multiple (LCM)

The LCM is the smallest positive number that is divisible by all given numbers. For example:

LCM of 4 and 6 is 12 (multiples: 4: 4,8,12... | 6: 6,12,18...)
LCM of 3, 4, and 5 is 60
LCM of 7 and 11 is 77 (product of primes)

Relationship Between GCF and LCM

For two numbers a and b, there's a special relationship:

GCF(a, b) × LCM(a, b) = a × b

For example: GCF(12, 18) × LCM(12, 18) = 6 × 36 = 216 = 12 × 18 ✓

Methods to Find GCF

Listing Method: List all factors and find the largest common one
Prime Factorization: Multiply common prime factors
Euclidean Algorithm: Efficient method using division
Division Method: Divide by common factors until no more exist

Applications

Simplifying Fractions: Divide numerator and denominator by GCF
Scheduling Problems: Finding when events coincide (LCM)
Cutting Materials: Finding maximum size that divides evenly (GCF)
Grouping Items: Arranging objects in equal groups
Music & Rhythm: Finding common beats and measures

Real-World Examples

Tiling a Floor: Use GCF to find largest tile size that fits perfectly
Traffic Lights: Use LCM to find when signals sync up
Recipe Scaling: Use GCF to reduce ingredient ratios
Bus Schedules: Use LCM to find when buses arrive together

Special Properties

GCF of any number with 1 is always 1
GCF(a, a) = a and LCM(a, a) = a
If two numbers are coprime (GCF = 1), their LCM is their product
GCF divides both the numbers and their difference