# Least Common Multiple

Least common multiple which is also mentioned as L.C.M. is the basics of Mathematics. This topic, we have learned in our primary classes. 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.

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.

## How to find Least Common Multiple?

As we have already discussed, the least common multiple is the smallest common factor or 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.

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.

## Least Common Multiple 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. = $\frac{L.C.M Of Numerator}{L.C.M Of Denominator}$

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

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