LCM and GCF: What They Are and How to Find Them
The least common multiple (LCM) and the greatest common factor (GCF) are two of the most useful tools in number theory. They appear in fraction calculations, scheduling problems, gear ratios, tiling puzzles and many real-world situations.
Despite sounding technical, both concepts are easy to understand once you see a few examples.
What Is the Greatest Common Factor?
The GCF of two or more numbers is the largest number that divides exactly into all of them without leaving a remainder. Also called the greatest common divisor (GCD) or highest common factor (HCF).
Example: What is the biggest number that goes evenly into both 12 and 18?
Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6. GCF = 6.
Method 1: Listing Factors
Write out all the factors of each number and find the largest one that appears in every list. Works well for small numbers.
Method 2: Prime Factorisation for GCF
Break each number into its prime factors. The GCF is the product of the prime factors they share.
Example: GCF of 36 and 60
36 = 2² × 3²
60 = 2² × 3 × 5
Shared prime factors: 2² and 3
GCF = 4 × 3 = 12
Method 3: Euclid's Algorithm
The most efficient method for large numbers. Divide the larger number by the smaller and find the remainder. Replace the larger number with the smaller and the smaller with the remainder. Repeat until the remainder is zero. The last non-zero remainder is the GCF.
Example: GCF of 252 and 105 Using Euclid's Algorithm
252 ÷ 105 = 2 remainder 42
105 ÷ 42 = 2 remainder 21
42 ÷ 21 = 2 remainder 0
GCF = 21
What Is the Least Common Multiple?
The LCM of two or more numbers is the smallest number that is a multiple of all of them — the smallest number they all divide into evenly.
Example: What is the smallest number that both 4 and 6 divide into evenly?
Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... First common multiple: 12. LCM = 12.
Method: Using the GCF to Find the LCM
There is a shortcut: LCM = (First number × Second number) / GCF
Example: LCM of 12 and 18 Using the GCF Shortcut
GCF of 12 and 18 = 6
LCM = (12 × 18) / 6 = 216 / 6
LCM = 36
Method: Prime Factorisation for LCM
Take the prime factorisation of each number. The LCM is the product of each prime factor at its highest power.
Example: LCM of 12 and 18 Using Prime Factorisation
12 = 2² × 3
18 = 2 × 3²
Take each prime at its highest power: 2² and 3²
LCM = 4 × 9 = 36
Where GCF and LCM Are Used
| Application | Which to use and why |
|---|---|
| Simplifying fractions | Divide numerator and denominator by their GCF |
| Adding fractions with different denominators | Find the LCM to use as the common denominator |
| Scheduling problems | When will two events coincide again? Find the LCM of their periods. |
| Gear ratios | GCF helps simplify gear ratios |
| Tiling patterns | LCM helps find when a pattern repeats |
Practical Example: Adding Fractions
Using LCM to Add 1/4 + 1/6
Find the common denominator: LCM of 4 and 6 = 12
1/4 = 3/12
1/6 = 2/12
1/4 + 1/6 = 3/12 + 2/12 = 5/12
Use the Fraction Calculator to handle this automatically.
Related Tools
- Fraction Calculator which uses LCM internally when adding and subtracting
- Scientific Calculator for prime factorisation and related calculations
- Ratio Calculator for simplifying and comparing ratios
- All Math Tools browse the full collection
Frequently Asked Questions
What is the difference between GCF and LCM?
The GCF is the largest number that divides into both numbers. The LCM is the smallest number that both numbers divide into. GCF makes numbers smaller (used in simplifying). LCM makes numbers larger (used in finding common denominators).
Is the GCF always smaller than both numbers?
Yes, the GCF is always less than or equal to the smaller of the two numbers. If two numbers share no common factors other than 1, their GCF is 1 and they are called coprime.
When is the LCM the same as multiplying the two numbers together?
When the two numbers are coprime (their GCF is 1). For example, the LCM of 7 and 9 is 63, which is 7 × 9, because they share no common factors.
How is Euclid's algorithm more efficient than listing factors?
Listing factors requires checking every number up to the smaller value. Euclid's algorithm reduces the problem rapidly with each step. For large numbers like 3,528 and 3,780, listing factors is tedious while Euclid's algorithm reaches the answer in just a few steps.