GCD Calculator
Find the greatest common divisor (GCD) of two or more numbers with step-by-step solutions.
Enter two or more numbers to find their Greatest Common Divisor
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Understanding GCD
A Comprehensive Guide to Greatest Common Divisor
The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF) or Highest Common Factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder.
Basic Concepts
Key Points:
- The GCD is always a positive integer
- GCD(a,b) = GCD(|a|,|b|) for any integers a and b
- GCD(a,0) = |a| for any non-zero integer a
- If a divides b, then GCD(a,b) = |a|
The Euclidean Algorithm
The Euclidean algorithm is an efficient method for computing the GCD of two numbers. It's based on the principle that the GCD of two numbers also divides their difference.
For numbers a and b:
GCD(a,b) = GCD(b, a mod b)
Steps:
- 1. Start with two numbers a and b
- 2. Divide a by b to get quotient q and remainder r
- 3. If r = 0, b is the GCD
- 4. Otherwise, set a = b and b = r, and repeat from step 2
Example: GCD(48,36)
48 = 36 × 1 + 12
36 = 12 × 3 + 0
Therefore, GCD(48,36) = 12
Properties of GCD
- Commutative: GCD(a,b) = GCD(b,a)
- Associative: GCD(a,GCD(b,c)) = GCD(GCD(a,b),c)
- Multiple Numbers: GCD can be found by repeatedly finding GCD of pairs
Real-Life Applications
Tiling a Floor
When tiling a rectangular floor with square tiles, the GCD of the length and width gives the largest possible size for the square tiles that will fit evenly.
Example: For a 48ft × 36ft room
GCD(48,36) = 12
Therefore, 12ft × 12ft is the largest square tile size that will fit evenly.
Gear Ratios
In mechanical engineering, GCD helps determine how often two gears return to their starting position.
Example: For gears with 48 and 36 teeth
GCD(48,36) = 12
The gears return to their starting position every 12 rotations.
Tips & Best Practices
Quick Tips:
- • Look for common factors first
- • Use prime factorization for larger numbers
- • Remember that GCD ≤ min(a,b)
- • Check if numbers are coprime (GCD = 1)
Watch Out For:
- • Forgetting to consider negative numbers
- • Mixing up GCD with LCM
- • Not checking all common factors
- • Assuming larger factors first