Least Common Multiple (LCM) - Sixth Grade Mathematics

The Least Common Multiple is the smallest positive whole number that is a multiple of two or more given numbers, forming the foundation for finding common denominators when adding and subtracting fractions.

Conceptual Foundation of Common Multiples

Every whole number has an infinite sequence of multiples. When comparing two or more numbers, their multiples eventually overlap at common values. The smallest of these shared positive values is the Least Common Multiple (LCM).

Multiples of 8:  8, 16, 24, 32, 40, 48, 56, 64, 72, ...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, ...

Shared Common Multiples: { 24, 48, 72, ... }
Least Common Multiple: 24

Methods for Calculating LCM

You can determine the LCM of two or more numbers using three reliable techniques: listing multiples, prime factorization, and continuous common division.

+-------------------------------------------------------------------------+
|                         THREE METHODS TO FIND LCM                       |
+-------------------+-----------------------+-----------------------------+
| Method            | Procedure             | Best Applied To             |
+-------------------+-----------------------+-----------------------------+
| Listing Multiples | Write consecutive     | Small numbers where LCM is  |
|                   | multiples until match | reached within 5 to 6 steps |
+-------------------+-----------------------+-----------------------------+
| Prime             | Decompose to primes;  | Standard multi-digit        |
| Factorization     | multiply HIGHEST power| numbers, algebraic terms    |
|                   | of ALL prime factors  |                             |
+-------------------+-----------------------+-----------------------------+
| Continuous        | Divide numbers        | Fast computation for two or |
| Division          | simultaneously by     | three numbers at once       |
|                   | common prime factors  |                             |
+-------------------+-----------------------+-----------------------------+

The Prime Factorization Method

To calculate the LCM using prime factorization: First, write the prime factorization of each number in exponential form. Second, identify all distinct prime bases that appear in any of the factorizations (even if a prime appears in only one number). Third, for each distinct prime base, choose the term with the highest exponent. Fourth, multiply these highest-power terms together.

Find LCM of 18 and 24:
18 = 2 * 3^2
24 = 2^3 * 3

All prime bases present: 2 and 3.
Highest power of 2: 2^3 (from 24)
Highest power of 3: 3^2 (from 18)

LCM(18, 24) = 2^3 * 3^2 = 8 * 9 = 72.

The Continuous Common Division Method

In continuous common division, write the numbers side by side on a line. Divide by any prime number that divides evenly into at least two of the numbers. If a number is not divisible, simply bring it down unchanged to the next line. Continue dividing until no two numbers share a common factor other than 1. Finally, multiply all the divisors along the left column and all the remaining quotients across the bottom row to obtain the LCM.

Find LCM of 12, 15, 20 using Common Division:
2 | 12  15  20
  +-----------
2 |  6  15  10   (15 brought down)
  +-----------
3 |  3  15   5   (5 brought down)
  +-----------
5 |  1   5   5   (1 brought down)
  +-----------
     1   1   1

Divisors on left: 2 * 2 * 3 * 5 = 60. Bottom row: 1 * 1 * 1.
LCM = 60.

The Fundamental Product Relationship Formula

For any two positive whole numbers a and b, there exists a direct mathematical connection between their product, their GCF, and their LCM.

           GCF(a, b) * LCM(a, b) = a * b

Rearranged formulas:
           LCM(a, b) = (a * b) / GCF(a, b)
           GCF(a, b) = (a * b) / LCM(a, b)

Applying the Relationship Formula

Consider the numbers 14 and 35. Their GCF is 7 (since 14 = 2 * 7 and 35 = 5 * 7). Instead of listing multiples of 35, apply the formula: LCM = (14 * 35) / 7. Simplify before multiplying: 14 / 7 = 2, and 2 * 35 = 70. This relationship saves substantial time when working with large numbers.

Exercises

Exercise 1. Find the Least Common Multiple of 15 and 20 using the listing multiples method.

Exercise 2. Use prime factorization to calculate the LCM of 36 and 54.

Exercise 3. Two numbers have a product of 1,800 and their Greatest Common Factor is 6. Calculate their Least Common Multiple using the product relationship formula.

Solutions

Solution to Exercise 1. List multiples of each number starting with the larger number: Multiples of 20: 20, 40, 60, 80, 100. Multiples of 15: 15, 30, 45, 60, 75. Comparing the lists, the first common multiple encountered is 60. Therefore, LCM(15, 20) = 60.

Solution to Exercise 2. Find the prime factorizations: 36 = 4 * 9 = 2^2 * 3^2. For 54: 54 = 2 * 27 = 2 * 3^3. Identify all distinct prime bases: 2 and 3. Select the highest exponent for each: the highest power of 2 is 2^2, and the highest power of 3 is 3^3. Multiply them: LCM = 2^2 * 3^3 = 4 * 27 = 108.

Solution to Exercise 3. Use the product formula: GCF(a, b) * LCM(a, b) = a * b. Substitute the given values: 6 * LCM = 1,800. Solve for LCM by dividing both sides by 6: LCM = 1,800 / 6 = 300. The Least Common Multiple of the two numbers is 300.