## Introduction - Least Common Multiple (LCM)

The Perfect Couple

Mona and Oliver are actually dating each other. First they went swimming and then a big pizza for everyone.

What are the characteristics of Mona and Oliver as a couple?

Mona and Oliver as a couple:

• likes listening to music
• likes to swim
• likes to eat pizza
• likes horses
• likes dogs
• plays on the piano
• plays the drums
What's the LCM of 12 and 980?

At the page prime factorization we have found the following disassemblies for the numbers 12 and 980:

• 2 × 2 × 3 = 12
• 2 × 2 × 5 × 7 × 7 = 980

Now, what is the Least Common Multiple (LCM) for these two numbers?

Proceed exactly the same as with our perfect couple Mona and Oliver - combine their characteristics: 2 × 2 × 3 × 5 × 7 × 7.

Therefore, the Least Common Multiple (LCM) of 12 and 980 is 2940.

## Importance of the Least Common Multiple (LCM)

It is worth to think briefly about the meaning of the term Least Common Multiple.

The LCM always refers to at least two numbers and represents a number that is a multiple of all these numbers. That's why it's called "Common Multiple".

One more Common Multiple of 12 and 980 is 5880.

However, it is not a question about any Common Multiple - we are looking for the least!

There is no least Common Multiple of 12 and 980 than 2940.

The LCM will play an important role for finding the Common Denominator, as well as when adding fractions and when subtracting fractions.

## Least Common Multiple - other terms

The Least Common Multiple (LCM) is also known as Lowest Common Multiple.

## Calculation of LCM through prime factorization

Find the Least Common Multiple (LCM) of a set of numbers by:

1. disassembling them into prime factors and
2. multiplying all the prime factors (all common prime factors only count as ones)

You can use a table as a sub-calculation:

• Insert the prime factors in the first row as headline
• Insert for each number in a separate line how often the corresponding prime factor occurs in the disassembly; insert the number itself into the last column.
• You obtain the prime factors of the Least Common Multiple (LCM) by writing the highest number of each prime factor into the last row.
• Finally calculate the Least Common Multiple (LCM) by multiplying its prime factors. Enter the result in the field at the bottom right.

## Example: Calculation of LCM through prime factorization

Example: Calculate the LCM of 297, 1386 and 396!

How to calculate the disassembly into prime numbers? Have a look at the page Calculation of the Greatest Common Divisor (GCD) and get a step-by-step explanation.

The result:

3 × 3 × 3 × 11 = 297
2 × 3 × 3 × 7 × 11 = 1386
2 × 2 × 3 × 3 × 11 = 396

After that, the table looks like this: In order to find the Least Common Multiple (LCM), we only have to multiply the prime factors of the 3 numbers (all common prime factors only count as ones).

The Least Common Multiple of 297, 1386 and 396 is:
2 × 2 × 3 × 3 × 3 × 7 × 11 = 8316.

Insert the highest number of each prime factor into the last row. Finally, calculate the LCM by multiplying its prime factors and insert the result in the field at the bottom right: 