Greatest Common Factor Calculator Find the GCF of two or more integers — with Euclidean steps, prime factors, and real-wo...
Greatest Common Factor Calculator
Find the GCF of two or more integers — with Euclidean steps, prime factors, and real-world applications.
Euclidean Algorithm (fastest)**:
$$ \gcd(a, b) = \gcd(b, a \bmod b) $$ Repeat until remainder = 0 → last non-zero remainder = GCF
**Example** (48, 180): 180 = 3 × 48 + 36 48 = 1 × 36 + 12 36 = 3 × 12 + 0 → **GCF = 12**
Prime Factorization**:
48 = 2⁴ × 3¹ 180 = 2² × 3² × 5¹ → GCF = 2min(4,2) × 3min(1,2) = 2² × 3 = **12**
✅ Pro Tip**: For two numbers: **GCF × LCM = a × b** → LCM(48,180) = (48×180)/12 = **720**
⚠️ Avoid these common errors:
- Confusing GCF with LCM** — GCF = largest *divisor*; LCM = smallest *multiple*
- Ignoring negatives** — gcd(−48, 180) = gcd(48, 180) = 12 (always positive)
- Zero edge cases** — gcd(a, 0) = |a|; gcd(0, 0) = 0 (by convention)
- Listing all factors for large numbers** — Euclidean is faster (e.g., gcd(10⁹, 10⁹+1) = 1 in 2 steps)
✅ Real-World Uses**:
- Fractions**: Simplify 48/180 → divide by 12 → **4/15**
- Tiling**: 120″ × 84″ room → largest square tile = **12″ × 12″**
- Ratios**: 24:36 → divide by 12 → **2:3**
- Coding**: `math.gcd()` in Python, `BigInteger.gcd()` in Java
| Numbers | GCF | Status | Reduced Ratio |
|---|---|---|---|
| 15, 25 | 5 | Composite | 3 : 5 |
| 17, 23 | 1 | Coprime | 17 : 23 |
| 32, 48 | 16 | Composite | 2 : 3 |
| 97, 101 | 1 | Coprime (primes) | 97 : 101 |
📉 Efficiency Comparison** (finding gcd(1,000,000, 1,000,001)):
- Listing factors: ~2,000 divisions
- Euclidean: **2 steps** (1,000,001 mod 1,000,000 = 1; 1,000,000 mod 1 = 0)
➡️ Two Numbers
“GCF of 48 and 180?” → **12** (Euclidean steps shown)
➡️ Three+ Numbers
“GCF of 24, 36, 60?” → gcd(gcd(24,36),60) = gcd(12,60) = **12**
➡️ Prime Factorization
“Factor 84 and 120” → see Venn diagram of common primes (2² × 3 = 12)
Note: Handles negatives (uses absolute values). Zero inputs follow mathematical conventions. All outputs integers ≥ 0.