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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{a} and b{b} we can find least common multiple.

It is denoted by LCM(a,b){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{a} and b{b} find prime factorization of a{a} and b{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=22⋅33{108}={{2}}^{{2}}\cdot{{3}}^{{3}} and 23⋅32{{2}}^{{3}}\cdot{{3}}^{{2}}.

108 72 Greater Factor
Factor 2 22{{2}}^{{2}} 23{{2}}^{{3}} 23{{2}}^{{3}}
Factor 3 33{{3}}^{{3}} 32{{3}}^{{2}} 33{{3}}^{{3}}

So, LCM(108,72)=23⋅33=8⋅27=216{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=24⋅32{144}={{2}}^{{4}}\cdot{{3}}^{{2}} and 54=21⋅33{54}={{2}}^{{1}}\cdot{{3}}^{{3}} we see that common factors are 2 and 3.

144 54 Greater Factor
Factor 2 24{{2}}^{{4}} 21{{2}}^{{1}} 24{{2}}^{{4}}
Factor 3 32{{3}}^{{2}} 33{{3}}^{{3}} 33{{3}}^{{3}}

Therefore, LCM(144,54)=24⋅33=432{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=22⋅33⋅5⋅7{3780}={{2}}^{{2}}\cdot{{3}}^{{3}}\cdot{5}\cdot{7} and 7056=24⋅32⋅72{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 50{{5}}^{{0}} because 50=1{{5}}^{{0}}={1}: 7056=24⋅32⋅50⋅72{7056}={{2}}^{{4}}\cdot{{3}}^{{2}}\cdot{{5}}^{{0}}\cdot{{7}}^{{2}}.

3780 7056 Greater Factor
Factor 2 22{{2}}^{{2}} 24{{2}}^{{4}} 24{{2}}^{{4}}
Factor 3 33{{3}}^{{3}} 32{{3}}^{{2}} 33{{3}}^{{3}}
Factor 5 51{{5}}^{{1}} 50{{5}}^{{0}} 51{{5}}^{{1}}
Factor 7 71{{7}}^{{1}} 72{{7}}^{{2}} 72{{7}}^{{2}}

So, LCM(3780,7056)=24⋅33⋅51⋅72=105840{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. GCD(a,b)⋅LCM(a,b)=ab{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 (GCD(a,b)=1)\left({G}{C}{D}{\left({a},{b}\right)}={1}\right) then LCM(a,b)=ab{L}{C}{M}{\left({a},{b}\right)}={a}{b}.