Introduction To LCM

The term $LCM$ means $Lowest\ Common\ Multiple$

Given two or more numbers, any number which is divisible by these two numbers is called a $Common\ Multiple$.

For example, if we take the number $6$ and $9$, then the common multiples for these two number could be $18$, $36$, $48$, $72$ and so on, since all these numbers are divisible by $6$ as well as $9$.

However, the smallest number which is divisible by the given numbers is called the $LCM$, which in our example would be $18$.

Can you think of any number smaller than $18$, which is divisible by both $6$ and $9$. There isn't any.

Now, we will see how to find the LCM of any two numbers. Let us go back to our example of $6$ and $9$. If we factorise both the numbers, we get:

$6 = 3 \times 2$, and

$9 = 3 \times 3$

We see that $3$ occurs once in $6$ and twice in $9$.

So we count $3$ once. To be divisible by $6$ our number needs one more $2$ and to be divisible by $9$ our number needs one more $3$. Therefore, the LCM of $6$ and $9$ is $3 \times 2 \times 3 = 18$.

So, to calculate the LCM of two or more numbers we first factorise the numbers and take all the factors that are common between the two. The multiply the result with the remaining factors of both numbers.

-----------book page break-----------

We will take slightly larger numbers to understand this better. Let us take the numbers $315$, and $450$.

We get:

$315 = 3 \times 3 \times 5 \times 7$, and, $450 = 2 \times 3 \times 3 \times 5 \times 5$

We can see that in the two given numbers the common factors are $3$, $3$ and $5$.

We will strike out the common numbers and write them separately. We get,

$\cancel{3} \times \cancel{3} \times \cancel{5} \times 7$ and $2 \times \cancel{3} \times \cancel{3} \times \cancel{5} \times 5$

We write the cancelled numbers as a part of our result.

$Result = 3 \times 3 \times 5$

Now we see that in the first number we have only $7$ left and in the second number we have only $2$ and $5$ remaining.

So, we multiply them with our result and get:

$Result = 3 \times 3 \times 5 \times 7 \times 2 \times 5$

$= 3150$

Hence, the LCM of $315$ and $450$ is $3150$