LCM (Least Common Multiple)

Introduction

LCM (Least Common Multiple or Lowest Common Multiple), they all are the same terms. We use them while finding out the number which is the lowest common multiple among many multiples of a number.

To see it in more detail how it is calculated, first of all, let’s review some general terms related to LCM.

Multiple

A multiple is a number that we can get from a number when multiplied by any positive integer.

Example

What are the multiples of 7?

We can say there are unlimited number of multiples of 7.

To be short, 14 is a multiple of 7.

Why?

\(\because\) here, 7 is the number

when 7 is multiplied by 2, the obtained number 14 is multiple 7.

How?

\({7} \times {2} = 14\)

Still, we can create more multiples of 7 by multiplying 7 with positive integers such as 3, 4, 5, 6, 7, 8….. and so on

\({7} \times {3} = 21\)

\({7} \times {4} = 28\)

\({7} \times {5} = 35\)

\({7} \times {6} = 42\)

\({7} \times {7} = 49\)

\({7} \times {8} = 56\)

So, here, 21, 28, 35, 42, 49, 56 are also multiples of 7.

Common Multiples

Common multiples are calculated for two or more than two numbers. They are the multiples which are common among the multiples of the given numbers.

Example

Let’s understand it by taking the two numbers as 2 and 4 and further, finding out their common multiples.

Step1: find out the multiples of 2 and 4 separately.

Multiples of 2:

\({2} \times {1} = 2\)

\({2} \times {2} = 4\)

\({2} \times {3} = 6\)

\({2} \times {4} = 8\)

\({2} \times {5} = 10\)

\({2} \times {6} = 12\)

The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.

Multiples of 4:

\({4} \times {1} = 4\)

\({4} \times {2} = 8\)

\({4} \times {3} = 12\)

\({4} \times {4} = 16\)

\({4} \times {5} = 20\)

\({4} \times {6} = 24\)

The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.

Step2: Find out the common multiples of 2 and 4.

Finally, we can say 4, 8, 12 are the common multiples of 2 and 4, because the multiples 4, 8 and 12 do exist for both numbers 2 and 4.

Least Common Multiple (LCM)

LCM (Least Common Multiple or Lowest Common Multiple) is the smallest number among the common multiples of given numbers.

Example

To understand LCM, lets reconsider the above example of common multiples, but now finding here the LCM of 2 and 4.

Find the LCM of 2 and 4.

So, what are the steps?

Step 1: Find out the multiples of 2.

Step 2: Find out the multiples of 4.

Step 3: Find out the common multiples of 2 and 4.

Step 4: Find out the smallest number among those common multiples, that will be the LCM of 2 and 4.

Step 1: Find out the multiples of 2.

\({2} \times {1} = 2\)

\({2} \times {2} = 4\)

\({2} \times {3} = 6\)

\({2} \times {4} = 8\)

\({2} \times {5} = 10\)

\({2} \times {6} = 12\)

The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.

Step 2: Find out the multiples of 4.

\({4} \times {1} = 4\)

\({4} \times {2} = 8\)

\({4} \times {3} = 12\)

\({4} \times {4} = 16\)

\({4} \times {5} = 20\)

\({4} \times {6} = 24\)

The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.

Step 3: Find out the common multiples of 2 and 4.

\(\therefore\) the common multiples are 4, 8 and 12.

Step4: Find out the smallest number among those common multiples, that would be the LCM of 2 and 4.

\(\therefore\) the smallest number among common multiples 4, 8 and 12 is 4.

Hence, LCM of 2 and 4 is 4.

Relationship between LCM and HCF

Product of LCM and HCF

Product of LCM and HCF of given numbers is equal to product of the numbers.

Formula

\(LCM \times HCF \; of \; given \; numbers = Product \; of \; the \; numbers \)