## Introduction - Least Common Multiple (LCM)

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

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:
*

- disassembling them into prime factors and
*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: