Prime factorisation breaks any whole number down into the prime numbers that multiply together to produce it. Every whole number greater than 1 has exactly one such breakdown, which makes it a reliable building block for other calculations like HCF and LCM.
A prime number is a whole number greater than 1 with no divisors other than 1 and itself (2, 3, 5, 7, 11...). Prime factorisation finds the unique set of primes that multiply together to give the original number. This calculator finds them by trial division: starting at 2, it divides the number repeatedly by each prime as long as it divides evenly, then moves to the next candidate. For example, 84 divides by 2 twice (84 → 42 → 21), then by 3 once (21 → 7), then 7 is itself prime — giving 84 = 2 × 2 × 3 × 7, or 2² × 3 × 7 in exponential form.
The highest common factor (HCF, also called the greatest common divisor or GCD) is the largest number that divides evenly into both numbers entered — useful for simplifying a fraction to its lowest terms, since dividing numerator and denominator by their HCF reduces it in one step. The lowest common multiple (LCM) is the smallest number that both numbers divide into evenly — this is what you need as a common denominator when adding fractions, and it also comes up in scheduling problems, such as working out when two repeating events (a bus every 12 minutes, another every 18 minutes) next coincide.
Writing a factorisation as a list (2 × 2 × 3 × 7) and as exponential form (2² × 3 × 7) both describe the same number, but the exponential form makes it much faster to compare two factorisations by eye — which is exactly what's needed to compute HCF (take the lowest shared power of each common prime) and LCM (take the highest power of every prime that appears in either number).