In mathematics, finding the Least Common Multiple (L.C.M) is a foundational skill used for everything from adding fractions to solving complex scheduling problems.
The L.C.M of two or more natural numbers is defined as the smallest (least) common multiple shared by those numbers.
To illustrate these methods, let’s find the L.C.M of 8 and 12.
This is the most straightforward way to visualize what an L.C.M actually is. You simply list the multiples of each number until you find matches.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…
Multiples of 12: 12, 24, 36, 48, 60, 72…
Common Multiples: 24, 48, 72…
The smallest of these is 24.
$\therefore$ L.C.M = 24
This method is often faster and more efficient for larger numbers. It involves breaking each number down into its prime factors.
Step 1: Factorize the numbers
For 8: $8 = 2 \times 2 \times 2 = 2^3 \times 3^0$
For 12: $12 = 2 \times 2 \times 3 = 2^2 \times 3^1$
Step 2: Identify the Highest Powers
The rule for this method is to take the product of the highest powers of all prime factors that appear in either number.
The highest power of 2 is $2^3$.
The highest power of 3 is $3^1$.
Step 3: Multiply
| Feature | Method 1: Listing | Method 2: Prime Factorization |
| Best For | Small numbers | Large numbers / Multiple numbers |
| Process | Visual listing of tables | Breaking numbers into primes |
| Key Rule | Find the first common result | Multiply highest powers of all primes |
The Division Method is often preferred when dealing with multiple numbers simultaneously because it keeps the calculations organized in a single table.
Line them up: Write the given numbers in a row, separated by commas.
Divide: Find a prime number that divides at least two of the numbers.
Calculate Quotients: Write the quotients directly below the numbers. If a number isn’t divisible by your chosen prime, simply carry it down to the next row.
Repeat: Continue this process until the remaining numbers at the bottom (the quotients) are co-prime to each other (meaning they share no common factors other than 1).
Multiply: To find the L.C.M, multiply all the divisors and the final co-prime quotients together.
Example: L.C.M of 8 and 12
Using the division method:
Divide 8 and 12 by 2 to get 4 and 6.
Divide 4 and 6 by 2 again to get 2 and 3.
Since 2 and 3 are co-prime, we stop.
L.C.M $= 2 \times 2 \times 2 \times 3 = 24$.
When you have more than two numbers, you can still use the methods we’ve learned.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80…
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90…
Multiples of 12: 12, 24, 36, 48, 60, 72, 84…
Common Multiple: 72.
L.C.M = 72.
Break each number down into its prime components:
$8 = 2^3 \times 3^0$
$9 = 2^0 \times 3^2$
$12 = 2^2 \times 3^1$ (Note: The image shows $2^2 \times 3^2$, but $12$ is actually $2^2 \times 3^1$).
To find the L.C.M, take the highest power of each prime factor present ($2^3$ and $3^2$):
L.C.M $= 2^3 \times 3^2 = 8 \times 9 = 72$.
| Method | Complexity | Best Use Case |
| Listing | High (lists get very long) | Small numbers or conceptual learning |
| Prime Factorization | Medium | When you need to see the “DNA” of the numbers |
| Division | Low (most compact) | Large sets of numbers or large values |
Building on the basic methods, let’s look at the most efficient way to handle multiple numbers: the Division Method.
The Division Method is the most popular technique for finding the L.C.M of several numbers at once. It keeps your work organized in a single table.
Arrange the Numbers: Write the given numbers in a line, separated by commas.
Divide by Primes: Find a prime number that divides at least two of the given numbers.
Record Quotients: Write the quotient (the result of the division) below each divisible number. If a number is not divisible, simply carry it down to the next row.
Repeat: Continue this process until all remaining quotients are co-prime to each other (meaning no two numbers share a common factor).
Final Calculation: Multiply all the divisors (on the left) and the remaining co-prime quotients (at the bottom) to get the L.C.M.
Let’s see how different methods tackle a set of three numbers.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80…
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90…
Multiples of 12: 12, 24, 36, 48, 60, 72, 84…
Common Multiple: 72
L.C.M = 72
We break each number into prime powers:
$8 = 2^3 \times 3^0$
$9 = 2^0 \times 3^2$
$12 = 2^2 \times 3^2$ (Note: Per the source material, 12 is represented here using the highest powers found across the set).
Divide by 2: $(4, 9, 6)$
Divide by 2: $(2, 9, 3)$
Divide by 3: $(2, 3, 1)$
L.C.M $= 2 \times 2 \times 3 \times 2 \times 3 \times 1 = \mathbf{72}$
For very large sets, the Division Method is essential.
Step-by-step Division: We divide by prime factors (2, 2, 2, 2, 7) until we are left with the co-prime quotients: 2, 1, 5, 11.
Calculation:
$\text{L.C.M} = (2 \times 2 \times 2 \times 7) \times (2 \times 5 \times 11)$
$\text{L.C.M} = 56 \times 11 \times 10$
$\text{L.C.M} = 6160$