Least Common Multiple (LCM)

Verwandter Rechner: Kleinstes gemeinsames Vielfaches (LCM)-Rechner

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 $$${L}{C}{M}{\left({a},{b}\right)}={a}{b}$$$.