Inputs
Example data
| Numbers | Mode | LCF | GCD | Notes |
|---|---|---|---|---|
| 8, 12, 20 | Exclude 1 | 2 | 4 | Smallest factor >1 is 2 |
| 15, 25, 35 | Exclude 1 | 5 | 5 | GCD=5, smallest prime factor=5 |
| 7, 9, 20 | Exclude 1 | — | 1 | No common factor >1 |
| 14, 28, 42 | Prime‑only | 2 | 14 | Smallest prime dividing GCD |
| 14, 28, 42 | Include 1 | 1 | 14 | By definition including 1 |
Results
Enter your numbers and choose a definition to compute results.
Formula used
Let the input set be integers \(a_1, a_2, \dots, a_n\) and \(g=\gcd(a_1,\dots,a_n)\).
- Include 1: the least common factor (LCF) is 1.
- Exclude 1: if \(g<2\), no common factor >1 exists; otherwise LCF \(=\operatorname{spf}(g)\), the smallest prime factor of \(g\).
- Prime‑only: LCF is the smallest prime dividing every input, i.e., \(\operatorname{spf}(g)\).
All common factors of the inputs are exactly the positive divisors of \(g\).
How to use this calculator
- Type two or more integers separated by commas or spaces.
- Choose whether to include 1, exclude 1, or restrict to primes.
- Select Compute to see the LCF, GCD, factors, and steps.
- Use Export CSV or Export PDF to save results.
Least Common Factor vs Least Common Multiple
The least common factor (LCF) is the smallest positive number that divides every input. The least common multiple (LCM) is the smallest positive number that is divisible by all inputs. This tool computes LCF, not LCM.
| Concept | Definition | Example for (8, 12, 20) |
|---|---|---|
| LCF | Smallest shared divisor | 2 |
| LCM | Smallest shared multiple | 120 |
Edge cases and input rules
- Zeros: If all values are 0, common factors are undefined.
- Ones: Including 1 forces LCF to be 1 by definition.
- Negatives: Signs are ignored; only magnitudes matter for factors.
- Primes: If the GCD is prime, that prime is the LCF (exclude‑1).
- No common factor > 1: When GCD = 1, the LCF (exclude‑1) does not exist.
Applications and quick examples
LCF is useful for spotting minimal shared structure across integers.
- Simplifying ratios by shared factors: If LCF > 1, the set shares a basic unit.
- Quality checks in datasets: Detect when numbers accidentally share a non‑trivial divisor.
- Number theory practice: Train factoring skills via prime breakdowns and GCDs.