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

\(2\times2\times2\times3=24\).