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# Lowest common multiple

A common multiple of two integers is a multiple of both numbers.

Common multiples of $\Tblue{3}$ and $\Tgreen{7}$ are $\Tred{21}$, $42$, $63$, etc.

$15$ is not a common multiple of $\Tblue{5}$ and $\Tgreen{9}$, because it is not a multiple of $9$.

Two integers have a unique lowest common multiple (LCM).

The LCM of $\Tblue{12}$ and $\Tgreen{15}$ is $\Tred{60}$ because $\Tred{60} = \Tblue{12} \times 5 = \Tgreen{15} \times 4$

The LCM of $\Tblue{8}$ and $\Tgreen{120}$ is $\Tred{120}$ because $\Tred{120} = \Tblue{8}\times 15$

The highest common factor of two numbers can be one of the numbers or the product of the numbers.

$$\LCM(\Tblue{6},\Tgreen{18}) = \Tred{18},\quad \LCM(\Tblue{6},\Tgreen{9}) = \Tred{18},\quad \LCM(\Tblue{20}, \Tgreen{9}) = \Tred{180}.$$

The product of the HCF and LCM is the product of the numbers.

$$\Tblue{m}\Tgreen{n} = \HCF(\Tblue{m},\Tgreen{n})\LCM(\Tblue{m},\Tgreen{n})$$

We have $\HCF(\Tblue{15}, \Tgreen{10}) = 5$, $\LCM(\Tblue{15}, \Tgreen{10}) = 30$ and $$150 = \Tblue{15}\times \Tgreen{10} = \HCF(\Tblue{15}, \Tgreen{10})\LCM(\Tblue{15},\Tgreen{10}).$$