1.5Lowest Common Multiple (LCM)

The lowest common multiple (LCM) is the smallest number that all of them divide into. Same factorisations, opposite rule: multiply every prime that appears anywhere, each at its largest index. The larger index is what every number needs to fit inside.

LCM rule

\displaystyle \text{LCM} = \text{product of } \textbf{all} \text{ prime factors, each with the } \textbf{highest} \text{ index}

The same two numbers and the same columns. The LCM takes every prime, each at its larger index.

Finding the LCM of two numbers

Find the LCM of $\displaystyle 40$ and $\displaystyle 84$ .

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\displaystyle 40

\displaystyle = 2^3 \times 5

\displaystyle 84

\displaystyle = 2^2 \times 3 \times 7

Take the highest index for each prime factor:

2^3

3^1

5^1

and

7^1

\displaystyle \text{LCM}

\displaystyle = 2^3 \times 3 \times 5 \times 7

\displaystyle = 8 \times 3 \times 5 \times 7

\displaystyle = 840

Finding the LCM of three numbers

Find the LCM of $\displaystyle 8$ , $\displaystyle 15$ and $\displaystyle 20$ .

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\displaystyle 8

\displaystyle = 2^3

\displaystyle 15

\displaystyle = 3 \times 5

\displaystyle 20

\displaystyle = 2^2 \times 5

Take the highest index for each prime:

2^3

3^1

and

5^1

\displaystyle \text{LCM}

\displaystyle = 2^3 \times 3 \times 5

\displaystyle = 8 \times 3 \times 5

\displaystyle = 120

Now you try

Find the LCM of $15$ , $20$ , and $30$ .

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\displaystyle 15

\displaystyle = 3 \times 5 \quad \scriptsize\textit{prime factorisation}

\displaystyle 20

\displaystyle = 2^{2} \times 5 \quad \scriptsize\textit{prime factorisation}

\displaystyle 30

\displaystyle = 2 \times 3 \times 5 \quad \scriptsize\textit{prime factorisation}

\displaystyle \text{LCM}

\displaystyle = 2^{2} \times 3 \times 5 \quad \scriptsize\textit{highest powers}

\displaystyle = 60

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