LCM Calculator
Calculate the Least Common Multiple (LCM) for up to 10 numbers. Study step-by-step derivations using Prime Factorization or Division Grid methods.
Enter 2 to 10 positive integers between 1 and 10,000.
Map fractions with denominators matching input list values.
Least Common Multiple (LCM)
24
Least Common Denominator (LCD) Alignments
| Number | Prime Factors | Expanded Form |
|---|---|---|
| 6 | 2 × 3 | 2 × 3 |
| 8 | 2<sup>3</sup> | 2 × 2 × 2 |
| 12 | 2<sup>2</sup> × 3 | 2 × 2 × 3 |
What is the Least Common Multiple (LCM)?
The **Least Common Multiple (LCM)** of two or more integers is the smallest positive integer that is evenly divisible by all the numbers in the set. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 can divide into without a remainder.
Finding common multiples is fundamental for working with fractions (to locate common denominators) and solving real-world scheduling or cycle-alignment puzzles.
Prime Factorization Method
To find the LCM of a set of numbers using the prime factorization method:
- Find the prime factors of each number (e.g. $12 = 2^2 \times 3^1$, and $18 = 2^1 \times 3^2$).
- For each distinct prime factor, identify the largest exponent (e.g. for base 2, it is $2^2$; for base 3, it is $3^2$).
- Multiply these highest powers together: $\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36$.
Division Grid Method (L-Method)
The **Grid Division Method** lists the numbers side-by-side in a table and divides them simultaneously by prime factors starting from 2. If a number is divisible, write down the quotient; otherwise, bring the number down unchanged. Continue this until all numbers are reduced to 1. The LCM is then found by multiplying all the prime divisors.