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