Least Common Multiple (LCM)

Find the LCM of two or more numbers using prime factorization, listing multiples, or GCF method.

Find the Least Common Multiple (LCM) of two or more numbers with step-by-step prime factorization. Also known as Lowest Common Multiple or Least Common Denominator (LCD).

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Finding the Least Common Multiple

This calculator finds the LCM of two or more numbers and shows the complete work using prime factorization. The LCM is the smallest positive integer that is evenly divisible by all numbers in the set.

Methods to Find LCM:

  1. Listing Multiples: List multiples of each number until you find the smallest common one
  2. Prime Factorization: Take the highest power of each prime factor from all numbers
  3. Using GCF: LCM(a,b) = (a × b) / GCF(a,b)
  4. Cake/Ladder Method: Divide by common primes repeatedly

Also Known As:

  • Least Common Multiple (LCM)
  • Lowest Common Multiple (LCM)
  • Least Common Divisor (LCD)
infoWhat is Least Common Multiple (LCM)?expand_more

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is evenly divisible by each of the given numbers. For example, LCM(2,3) = 6 and LCM(6,10) = 30.

The LCM is essential for adding and subtracting fractions with different denominators, scheduling problems (finding when events coincide), and any situation involving synchronization of repeating cycles.

Properties of LCM: The LCM is associative (LCM(a,b) = LCM(b,a)), commutative (LCM(a,b,c) = LCM(LCM(a,b),c)), and related to GCF by the formula LCM(a,b) × GCF(a,b) = a × b.

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Prime Factorization Method:

1. Find the prime factorization of each number

2. List all prime numbers found with their highest powers

3. Multiply these together to get the LCM

Using GCF Formula:

LCM(a, b) = (a × b) / GCF(a, b)

Relationship with GCF:

LCM(a, b) × GCF(a, b) = |a × b|

This relationship helps verify calculations

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Example 1: LCM(6, 7, 21) by Listing Multiples

Multiples of 6: 6, 12, 18, 24, 30, 36, 42...

Multiples of 7: 7, 14, 21, 28, 35, 42...

Multiples of 21: 21, 42...

LCM = 42 (smallest common multiple)

Example 2: LCM(12, 18, 30) by Prime Factorization

12 = 2² × 3

18 = 2 × 3²

30 = 2 × 3 × 5

Highest powers: 2², 3², 5¹

LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

Example 3: LCM(6, 10) using GCF

First find GCF(6, 10) = 2

Apply formula: LCM = (6 × 10) / 2

LCM = 60 / 2 = 30

quizFAQexpand_more
What is the difference between LCM and GCF?expand_more
LCM (Least Common Multiple) is the smallest number that all given numbers divide into evenly, while GCF (Greatest Common Factor) is the largest number that divides all given numbers evenly. They're related by: LCM(a,b) × GCF(a,b) = a × b.
What is the LCM of two prime numbers?expand_more
The LCM of two different prime numbers is simply their product. For example, LCM(3, 7) = 21, because prime numbers share no common factors other than 1, so their GCF is 1.
How is LCM used with fractions?expand_more
When adding or subtracting fractions with different denominators, you need to find a common denominator. The LCM of the denominators (also called LCD - Least Common Denominator) is the smallest common denominator you can use.
What if one number is a multiple of another?expand_more
If one number is already a multiple of the other, the LCM is simply the larger number. For example, LCM(4, 12) = 12, because 12 is already divisible by 4.
Which method is best for finding LCM?expand_more
For small numbers, listing multiples works well. For larger numbers, prime factorization or the GCF formula is more efficient. The cake/ladder method is popular for finding LCM of multiple numbers quickly.