# 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.**

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