LCM-Least Common Multiple

The definition of LCM states- Least common multiple is a method to find the smallest common multiple between any two or more numbers. Basically, the common multiple is a number which is a multiple of two or more numbers. L.C.M is used to determine the least common factor or multiple of any two or more given integers. For example, L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80, where 80 is the smallest common multiple for numbers 16 and 20. Now, if we consider the multiples of 16 (16,32,48,64,80,..) and 20 (20,40,60,80,..) we can see that the first common multiple for both the numbers is 80. This proves the method of LCM as correct.

Along with the least common multiple, you must have heard about the highest common factor, H.C.F., which is used to derive the highest common multiple factors of any two or more given integers. It is also called as Greatest Common Divisor. For example, the H.C.F. of 2,6,8 is 2, because all the three numbers can be divided with the highest number 2 commonly. H.C.F. and L.C.M. and both have equal importance in Maths.

In this article, we will learn how to find lcm for any two or more given integers with examples.

Also, read:

How to find LCM?

As we have already discussed, the least common multiple is the smallest common multiple for any two or more given numbers.

A multiple of a number we get, when we multiply a number with another number. Like 4 is a multiple of 2, as we multiply 2 with 2, we get 4. Similarly, in the case of maths table, you can see the multiples of a number when we multiply them from 1,2,3,4,5,6 and so on but not with zero. Now, if we have to find the common multiple of two or more numbers, then we have to write all the multiples for the given numbers. Say for example, if there are two numbers 4 and 6, then how to find the common multiple between them?

Let us write multiples of 4 and 6 first,

4 : 4,8,12,16,20,24,28,…..

6: 6,12,18,24,30,36,42…..

From the above two expressions you can see, 4 and 6 have common multiples as 12 and 24. They may have more common multiple if we go beyond. Now, the smallest or least common multiple for 4 and 6 you can see here is 12. Therefore, 12 is the L.C.M of 4 and 6. Also, find to learn LCM of two numbers here.

Now, let us take an example of 3 numbers.

Example: Find the L.C.M 4,6 and 12.

Solution: First write the common multiples of all the three numbers.

4 : 4,8,12,16,20,24,28,…..

6: 6,12,18,24,30,36,42…..

12: 12,24,36,48,60,72,….

From the above-given multiples of 4, 6 and 12, you can see, 12 is the smallest common multiple.

Therefore, L.C.M. of 4, 6 and 12 is 12.

LCM Formula

Let a and b are two given integers. We can write the formula for L.C.M. on the basis of the greatest common divisor(gcd) as mentioned below.

L.C.M. (a,b) = \(\frac{a * b}{gcd(a,b)}\)

This is the formula for two integers. But for fractions, the formula of L.C.M. becomes;

L.C.M. = \(\frac{L.C.M Of Numerator}{L.C.M Of Denominator}\)

LCM Example

Example: Find L.C.M. of 10 and 20.

Solution: We know, for given two integers a and b,

L.C.M. (a,b) = \(\frac{a * b}{gcd(a,b)}\)

Therefore, L.C.M. (10,20) = \(\frac{10 * 20}{gcd(10,20)}\)

The greatest common divisor for 10 and 20 is 10.

Thus, L.C.M. (10,20) = \(\frac{200}{gcd(10)}\)

L.C.M. (10,20) = 20

LCM Methods

By Finding the Multiples:

The method to find the least common multiple of any given numbers is first to write down the multiples of individual numbers and then find the first common multiple between them. Suppose, there are two number 11 and 33. Then the multiples of 11 and 33 can be written as;

Multiples of 11 = 11, 22, 33, 44, 55, ….

Multiples of 33 = 33, 66, 99, ….

We can see, the first common multiple or the least common multiple for both the numbers is 33. Hence the LCM (11, 33) = 33.

By Prime Factorisation:

Another method to find the LCM of the given numbers is by prime factorization. Suppose, there are three numbers 12, 16 and 24. Let us write the prime factors of all three numbers individually.

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

24 = 2 x 2 x 2 x 3

Now writing the prime factors of all the three numbers together, we get;

12 x 16 x 24 = 2 x 2 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Now pairing the common prime factors we get the LCM. Hence, there are four pairs of 2 and one pair of 3. So the LCM of 12, 16 and 24 will be;

LCM (12, 16, 24) = 2 x 2 x 2 x 2 x 3 = 48

LCM Tree

The Least common multiple trees can be formed by using the prime factorisation method. Suppose there are two numbers 60 and 282. Then, first let us write the prime factors of these two numbers, such as;

60 = 6 x 10 = 2 x 3 x 2 x 5

282 = 2 x 141 = 2 x 3 x 47

Now let us represent the above prime factorization using a tree.

LCM tree

From the above tree diagram, we can take the pair of common factors and unique factors from the branches of both the numbers and multiply them as a whole to get the LCM. Therefore,

LCM (60, 282) = 2 x 2 x 3 x 5 x 47 = 2820

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