One number tells you the biggest thing two numbers can be evenly split into. The other tells you the soonest they'll ever line up again. Together, they're the payoff for everything you've learned about factors and multiples.
HCF and LCM take everything from our Factors & Multiples article and put it to work comparing two numbers at once — one by finding what they share, the other by finding where they'll next meet.
The Highest Common Factor (HCF) — also called the Greatest Common Divisor (GCD) — is the largest number that divides evenly into both of two given numbers.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12 — the highest of these is 12.
Break both numbers into prime factors, then multiply together only the primes they share, using the lowest power each prime appears with in either number.
18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
Shared primes: one 2 (the smaller count between the two) and one 3 (the smaller count between the two). HCF = 2 × 3 = 6.
Divide the larger number by the smaller, note the remainder, then repeat using the smaller number and that remainder — over and over — until the remainder hits zero. The last non-zero divisor is the HCF.
48 = 2 × 18 + 12
18 = 1 × 12 + 6
12 = 2 × 6 + 0
The remainder just hit 0, so the HCF is the last divisor used: 6.
This method is especially useful for large numbers, since it avoids listing out every factor.
The Lowest Common Multiple (LCM) is the smallest number that both of two given numbers divide into evenly — the earliest point where their multiple lists meet.
Multiples of 12: 12, 24, 36, 48...
Multiples of 18: 18, 36, 54...
The first shared value is 36.
Break both numbers into prime factors, then multiply together every prime that appears in either number, using the highest power it appears with.
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
Take the highest count of each prime: three 2's (from 24) and two 3's (from 36). LCM = 2×2×2×3×3 = 72.
For any two numbers, their HCF multiplied by their LCM always equals the two numbers multiplied together. This makes finding one instantly, once you already know the other.
For 24 and 36: HCF = 12, and 24 × 36 = 864. So LCM = 864 ÷ 12 = 72 — matching what the prime factorization method found directly.
Mixing up "highest" and "lowest" is the single most common HCF/LCM mistake.
The fix: HCF uses the lowest shared power of each common prime (only counting primes both numbers have). LCM uses the highest power of every prime that appears in either number. The names are the clue: Highest Common Factor still means taking the smaller shared amount; Lowest Common Multiple still means reaching the bigger combined result.
Two numbers are co-prime — a concept covered in our Number Systems article — exactly when their HCF is 1, meaning they share no common factor at all besides 1.
8 and 15: factors of 8 are 1,2,4,8; factors of 15 are 1,3,5,15. The only shared factor is 1, so HCF(8,15) = 1 — confirming 8 and 15 are co-prime. Their LCM is simply 8×15 = 120, since co-prime numbers always have an LCM equal to their straightforward product.
Tap any pair of numbers below to see their factors, common factors, HCF, and LCM all at once.
Tap a pair above to see the full breakdown.
Cutting two ribbons, 24cm and 36cm long, into the largest possible equal pieces with none wasted uses HCF: pieces of 12cm each. Two traffic lights that turn green together, one every 12 seconds and the other every 18 seconds, will next turn green together using LCM: every 36 seconds.
The Euclidean algorithm for finding HCF, described in our division method above, is named after the Greek mathematician Euclid, who documented it over 2,300 years ago. It's still considered one of the oldest algorithms still in common use today, including inside modern computer science.
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HCF is exactly how fractions get simplified to lowest terms — dividing numerator and denominator by their HCF. LCM is exactly how fractions with different denominators get added or subtracted — finding a common denominator is finding an LCM. Beyond fractions, HCF and LCM show up anywhere two repeating patterns or divisible groups need to be compared: scheduling, packaging, and construction all lean on these two ideas constantly.
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