Euclid’s algorithm and shared prime factors
The greatest common factor (GCF), greatest common divisor (GCD), and highest common factor (HCF) name the same value. Euclid’s algorithm repeatedly replaces a pair with its divisor and remainder until the remainder is zero. The final non-zero divisor is the GCF.
For more than two inputs, find the GCF of the first pair and combine that result with each remaining input. In prime factorization, the same answer comes from keeping only shared primes, each at its lowest exponent.
Example: reducing 48 : 180
180 = 3 × 48 + 36; 48 = 1 × 36 + 12; 36 = 3 × 12 + 0. The GCF is 12. Dividing both terms by 12 reduces 48 : 180 to 4 : 15. The common positive factors are 1, 2, 3, 4, 6 and 12.
Input conventions
This page accepts 2–100 non-zero integers, each within ±10¹², and uses their absolute values. Negative signs do not change the positive GCF. Decimals and invalid tokens are rejected rather than omitted.
The optional LCM is reported only when it fits safe integer precision. That limit does not prevent a valid GCF from being shown.