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Common GCD Values Quick Reference
- GCD(8, 12) = 4
- GCD(15, 25) = 5
- GCD(100, 75) = 25
- GCD(17, 13) = 1 (both prime, always coprime)
- GCD(n, n) = n (a number's GCD with itself is itself)
- GCD(n, 1) = 1 (any number and 1 are coprime)
- GCD(0, n) = n (GCD with zero is the other number)
When to Use GCD
- Simplifying fractions to lowest terms
- Finding common denominators (via LCM)
- Checking if two numbers are coprime (for RSA key generation)
- Solving tiling, grid, and partitioning problems
- Synchronizing repeating events or cycles
- Reducing ratios (e.g., aspect ratios like 1920:1080 → 16:9)
Examples
Example 1: Basic GCD of Two Numbers
Find the GCD of 48 and 18:
Input: 48, 18
Euclidean Algorithm Steps:
48 = 2 × 18 + 12 → remainder 12
18 = 1 × 12 + 6 → remainder 6
12 = 2 × 6 + 0 → remainder 0 (stop)
GCD(48, 18) = 6
The last non-zero remainder is the GCD. So 6 is the largest number that divides both 48 and 18 evenly.
Example 2: Simplifying Fractions
The most common everyday use of GCD is reducing fractions to their simplest form:
Fraction: 48/18
Step 1: Find GCD(48, 18) = 6
Step 2: Divide numerator and denominator by GCD
48 ÷ 6 = 8
18 ÷ 6 = 3
Simplified fraction: 8/3
More examples:
36/48 → GCD = 12 → 3/4
100/75 → GCD = 25 → 4/3
56/98 → GCD = 14 → 4/7
120/90 → GCD = 30 → 4/3
Example 3: GCD of Three or More Numbers
Find the GCD of 60, 90, and 120:
Step 1: GCD(60, 90)
90 = 1 × 60 + 30
60 = 2 × 30 + 0
GCD(60, 90) = 30
Step 2: GCD(30, 120)
120 = 4 × 30 + 0
GCD(30, 120) = 30
GCD(60, 90, 120) = 30
The GCD of multiple numbers is found by chaining pairwise GCD calculations.