# Least Common Multiple (LCM)

## Related Calculator: Least Common Multiple (LCM) Calculator

Suppose we are given two numbers 18 and 24.

Let's find some of their multiples.

18: 18,36,54,72,90,108,126,144,...

24: 24,48,72,96,120,144,168,...

As can be seen some factors are same for both numbers (they are in bold: 72 and 144). These numbers are called common multiples of 18 and 24.

The smallest of common multiples (in bold red) is called the least common multiple.

For any integer numbers a and b we can find least common multiple.

It is denoted by LCM(a,b) (short for the Least Common Multiple).

Let's see how to find least common multiple.

To Find Least Common Multiple of a and b find prime factorization of a and b and then take product of common factors taking each of them with greatest exponent.

Example 1 . Find LCM(108,72).

Find prime factorization: 108=2^2*3^3 and 2^3*3^2.

 108 72 Greater Factor Factor 2 2^2 2^3 2^3 Factor 3 3^3 3^2 3^3

So, LCM(108,72)=2^3*3^3=8*27=216.

Next example.

Example 2 . Find LCM(144,54).

Since 144=2^4*3^2 and 54=2^1*3^3 we see that common factors are 2 and 3.

 144 54 Greater Factor Factor 2 2^4 2^1 2^4 Factor 3 3^2 3^3 3^3

Therefore, LCM(144,54)=2^4*3^3=432.

Next example.

Example 3. Find LCM(3780,7056).

Find prime factorization: 3780=2^2*3^3*5*7 and 7056=2^4*3^2*7^2.

You can see that 7056 doesn't have 5 as factor, while 3780 has.

We can write in prime factorization of 7056 factor 5^0 because 5^0=1: 7056=2^4*3^2*5^0*7^2.

 3780 7056 Greater Factor Factor 2 2^2 2^4 2^4 Factor 3 3^3 3^2 3^3 Factor 5 5^1 5^0 5^1  Factor 7 7^1 7^2 7^2

So, LCM(3780,7056)=2^4*3^3*5^1*7^2=105840.

Now, take pen and paper and do following exercises.

Exercise 1. Find LCM(45,375).

Next exercise.

Exercise 2. Find LCM(63,450).

Fact. GCD(a,b)*LCM(a,b)=ab.
In particular, it means that if a and b are relatively prime (GCD(a,b)=1) then LCM(a,b)=ab.