Least Common Multiple (LCM)
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 $$${L}{C}{M}{\left({a},{b}\right)}$$$ (short for the Least Common Multiple).
Let's see how to find least common multiple.
To find the 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: $$${108}={{2}}^{{2}}\cdot{{3}}^{{3}}$$$ and $$${{2}}^{{3}}\cdot{{3}}^{{2}}$$$.
| 108 | 72 | Greater Factor | |
| Factor 2 | $$${{2}}^{{2}}$$$ | $$${{2}}^{{3}}$$$ | $$${{2}}^{{3}}$$$ |
| Factor 3 | $$${{3}}^{{3}}$$$ | $$${{3}}^{{2}}$$$ | $$${{3}}^{{3}}$$$ |
So, $$${L}{C}{M}{\left({108},{72}\right)}={{2}}^{{3}}\cdot{{3}}^{{3}}={8}\cdot{27}={216}$$$.
Next example.
Example 2. Find LCM(144,54).
Since $$${144}={{2}}^{{4}}\cdot{{3}}^{{2}}$$$ and $$${54}={{2}}^{{1}}\cdot{{3}}^{{3}}$$$ 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, $$${L}{C}{M}{\left({144},{54}\right)}={{2}}^{{4}}\cdot{{3}}^{{3}}={432}$$$.
Next example.
Example 3. Find LCM(3780,7056).
Find prime factorization: $$${3780}={{2}}^{{2}}\cdot{{3}}^{{3}}\cdot{5}\cdot{7}$$$ and $$${7056}={{2}}^{{4}}\cdot{{3}}^{{2}}\cdot{{7}}^{{2}}$$$.
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 $$${{5}}^{{0}}={1}$$$: $$${7056}={{2}}^{{4}}\cdot{{3}}^{{2}}\cdot{{5}}^{{0}}\cdot{{7}}^{{2}}$$$.
| 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, $$${L}{C}{M}{\left({3780},{7056}\right)}={{2}}^{{4}}\cdot{{3}}^{{3}}\cdot{{5}}^{{1}}\cdot{{7}}^{{2}}={105840}$$$.
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. $$${G}{C}{D}{\left({a},{b}\right)}\cdot{L}{C}{M}{\left({a},{b}\right)}={a}{b}$$$.
In particular, it means that if a and b are relatively prime $$$\left({G}{C}{D}{\left({a},{b}\right)}={1}\right)$$$ then $$${L}{C}{M}{\left({a},{b}\right)}={a}{b}$$$.