 ## Multiples, Common Multiples and Lowest Common Multiple (LCM)

 2.2 Multiples, Common Multiples and Lowest Common Multiple (LCM)

 Multiples The product of a number multiplied by a given number.

 Example $$9\times1=9 \\9\times2=18 \\9\times3=27 \\$$ $$9$$ is multiplied by $$1,2,3,..$$ will produce $$9,18,27,..$$. Thus, $$9,18,27,..$$ is the multiples of $$9$$.

 Common Multiples A number that is a multiple of two or more numbers.

 Example Is $$24$$ the common multiple of $$6$$ and $$8$$? $$24\div6=4 \\24\div8=3 \\$$ $$24$$ can be divided completely by $$6$$ and $$8$$. Thus, $$24$$ is the common multiple of $$6$$ and $$8$$.

 Lowest Common Multiple (LCM) The smallest common multiple obtained and it is the first common multiple if listed.

Solution Methods
 Listing the common multiples: (i) Determine the LCM of $$2$$ and $$3$$. Multiples of $$2: 2,4,6,8,..$$ Multiples of $$3: 3, 6, 9,..$$ Thus, the lowest common multiple of $$2$$ and $$3$$ is $$6$$. ​​ Repeated division: (ii) Determine the LCM of $$3,6$$ and $$9$$. $$\begin{array}{c} 3\\2\\3 \\\phantom{-} \end{array} \begin{array}{|c} \quad3,\,6,\,9\quad\\ \hline \quad1,\,2,\,3\quad\\ \hline \quad1,\,1,\,3\quad\\ \hline \quad1,\,1,\,1\quad\\ \end{array} \begin{array}{c}\end{array}\\\\$$ LCM of $$3,6$$ and $$9$$ is $$3\times2\times3 = 18$$. Prime factorisation: (iii) Determine the LCM of $$3,8$$ and $$12$$. \begin{aligned} 3&=\quad\quad\quad\quad\quad\,3 \\8&=2\times2\times2 \\12&=\quad\,\,\,\,\,2\times2\times3 \end{aligned}\\\\ LCM of $$3,8$$ and $$12$$ is $$2\times2\times2\times3=24$$.