CalcCalcolatrice di MCD e MCM
Ultimo aggiornamento: 2026-05-09
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| a | b | |
|---|---|---|
| Caso basico | 4.8 | 7.2 |
| Caso tipico | 8.4 | 12.6 |
| Caso medio | 12.0 | 18.0 |
| Caso avanzado | 18.0 | 27.0 |
| Caso extremo | 30.0 | 45.0 |
GCF & LCM Calculator: greatest common factor and least common multiple
This calculator finds the greatest common factor (GCF) and least common multiple (LCM) of two or more integers, fundamental tools for simplifying fractions and operating with different denominators.
GCF and LCM formulas
For two numbers a and b:
- GCF: computed using Euclid's algorithm, which repeatedly divides the larger by the smaller until the remainder is 0.
- LCM: obtained as LCM(a, b) = |a × b| / GCF(a, b).
The GCF is the largest number that divides both exactly. The LCM is the smallest number that is a multiple of both.
Example 1: GCF and LCM of 12 and 18
Problem: Find the GCF and LCM of 12 and 18.
- GCF by Euclid:
- 18 = 12 × 1 + 6; 12 = 6 × 2 + 0 → GCF = 6.
- LCM:
- LCM = |12 × 18| / 6 = 216 / 6 = 36.
Answer: GCF(12, 18) = 6, LCM(12, 18) = 36.
Example 2: GCF and LCM of 24 and 36
Problem: Find the GCF and LCM of 24 and 36.
- GCF by Euclid:
- 36 = 24 × 1 + 12; 24 = 12 × 2 + 0 → GCF = 12.
- LCM:
- LCM = |24 × 36| / 12 = 864 / 12 = 72.
Answer: GCF(24, 36) = 12, LCM(24, 36) = 72.
Usi comuni
- Simplifying fractions by dividing numerator and denominator by the GCF.
- Finding common denominators for adding or subtracting fractions using the LCM.
- Solving divisibility problems in arithmetic and number theory.
- Planning periodic events that coincide (synchronization problems).
- Factoring polynomials in algebra using the GCF of coefficients.
- Optimizing groupings and distributions in practical problems.
Common mistakes when computing GCF and LCM
- Confusing GCF with LCM: the GCF is less than or equal to the numbers, the LCM is greater than or equal.
- Not using absolute value when computing the product for LCM.
- Trying to list all divisors instead of using Euclid's algorithm.
- Applying the LCM formula without first computing the GCF correctly.
Consiglio professionale
Euclid's algorithm is extremely efficient: even for numbers with millions of digits, it converges in very few steps. It is much faster than listing all divisors or prime factors.
If two numbers share no common factors, their GCF is 1 and their LCM is simply their product.
Yes. Compute the GCF of two numbers and then use that result with the next: GCF(a, b, c) = GCF(GCF(a, b), c).
Yes. The GCF is always positive and so is the LCM. Absolute values of the numbers are used in the calculations.
For two numbers: GCF(a, b) × LCM(a, b) = |a × b|. This relationship lets you compute one if you know the other.