LCM and GCD are the two fundamental relationships between numbers that show up whenever fractions, ratios or repeating intervals are involved. Each has a clear purpose: GCD tells you the largest factor in common; LCM tells you the smallest common multiple.
GCD (Greatest Common Divisor / HCF)
The largest number that divides exactly into all given numbers.
Method 1 — List factors: GCD of 12 and 18:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- GCD: 6
Method 2 — Euclidean algorithm (faster for larger numbers): GCD(48, 18): 48 = 2 × 18 + 12 → GCD(18, 12) → 18 = 1 × 12 + 6 → GCD(12, 6) → 12 = 2 × 6 + 0 → GCD = 6
LCM (Lowest Common Multiple)
The smallest number that all given numbers divide into exactly.
Method 1 — List multiples: LCM of 4 and 6:
- Multiples of 4: 4, 8, 12, 16...
- Multiples of 6: 6, 12, 18...
- First common: 12
Method 2 — Using GCD (easiest once you have GCD): LCM(a, b) = (a × b) ÷ GCD(a, b)
LCM(12, 18) = (12 × 18) ÷ 6 = 216 ÷ 6 = 36
Practical Uses
| Situation | Which to use |
|---|---|
| Simplifying a fraction (12/18) | GCD — divide by it to simplify |
| Adding 1/4 + 1/6 (finding common denominator) | LCM — gives the denominator |
| Dividing 24 items into equal groups | GCD — how many per group |
| Two events repeating every 8 and 12 minutes; when do they coincide? | LCM — they coincide every 24 minutes |
Three or More Numbers
Apply GCD or LCM pairwise: GCD(12, 18, 24) = GCD(GCD(12,18), 24) = GCD(6, 24) = 6 LCM(4, 6, 8) = LCM(LCM(4,6), 8) = LCM(12, 8) = 24
Use LCM & GCD Calculator for quick results. For fraction calculations that use LCM for the denominator, see how to add and simplify fractions.
GCD and LCM in Fraction Work
The two appear together most naturally when working with fractions. To add 3/8 + 5/12: find LCM(8,12) = 24, convert both fractions to /24, then add. To simplify the result: find GCD of numerator and denominator, divide both by it. This pairing — LCM for finding common denominators, GCD for simplifying results — covers most of what you need for fraction arithmetic. Both LCM & GCD Calculator and Fraction Calculator handle these automatically.
GCD and Simplifying Ratios
Ratios simplify the same way fractions do — divide both parts by their GCD. A ratio of 18:24 simplifies to 3:4 (GCD = 6). This comes up in recipe scaling, material mixing and profit splits where the raw numbers are large but the relationship is simple. LCM & GCD Calculator computes GCD for any pair of numbers, and Ratio Calculator applies it directly to simplify ratios.
