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`.