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).

Answer: 1125.

Next exercise.

Exercise 2. Find LCM(63,450).

Answer: 3150.

Last one.

Exercise 3. Find LCM(13,45).

Answer: 585.

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`.