1.3 The least common multiple

We can visualize common multiples using the number line.



Notice that the common multiples can be divided by both whole numbers.

 9 and 12

1. $\begin{array}{lllllll}9\hfill & =\hfill & 3·3\hfill & =\hfill & 3^2\hfill & \hfill & \hfill \\ 12\hfill & =\hfill & 2·6\hfill & =\hfill & 2·2·3\hfill & =\hfill & 2^2·3\hfill \end{array}$
2. The bases that appear in the prime factorizations are 2 and 3.
3. The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2 and 2 (or $2^2$ from 12, and $3^2$ from 9).
4. The LCM is the product of these numbers.
   
   $\text{LCM\hspace{0.17em}}=2^2·3^2=4·9=36$

 90 and 630

1. $\begin{array}{lllllllll}90\hfill & =\hfill & 2·45\hfill & =\hfill & 2·3·15\hfill & =\hfill & 2·3·3·5\hfill & =\hfill & 2·3^2·5\hfill \\ 630\hfill & =\hfill & 2·315\hfill & =\hfill & 2·3·105\hfill & =\hfill & 2·3·3·35\hfill & =\hfill & 2·3·3·5·7\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & =\hfill & 2·3^2·5·7\hfill \end{array}$
2. The bases that appear in the prime factorizations are 2, 3, 5, and 7.
3. The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1.
   
   $\begin{array}{ll}{2}^1\hfill & \text{from\hspace{0.17em}either\hspace{0.17em}9}0\text{\hspace{0.17em}or\hspace{0.17em}63}0\hfill \\ 3^2\hfill & \text{from\hspace{0.17em}either\hspace{0.17em}9}0\text{\hspace{0.17em}or\hspace{0.17em}63}0\hfill \\ 5^1\hfill & \text{from\hspace{0.17em}either\hspace{0.17em}9}0\text{\hspace{0.17em}or\hspace{0.17em}63}0\hfill \\ 7^1\hfill & \text{from\hspace{0.17em}63}0\hfill \end{array}$
4. The LCM is the product of these numbers.
   
   $\text{LCM\hspace{0.17em}}=2·3^2·5·7=2·9·5·7=630$

 33, 110, and 484

1. $\begin{array}{lllll}33\hfill & =\hfill & 3·11\hfill & \hfill & \hfill \\ 110\hfill & =\hfill & 2·55\hfill & =\hfill & 2·5·11\hfill \\ 484\hfill & =\hfill & 2·242\hfill & =\hfill & 2·2·121=2·2·11·11=2^2·{11}^2\hfill \end{array}$
2. The bases that appear in the prime factorizations are 2, 3, 5, and 11.
3. The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2.
   
   $\begin{array}{ll}{2}^2\hfill & \text{from\hspace{0.17em}}484\hfill \\ 3^1\hfill & \text{from\hspace{0.17em}}33\hfill \\ 5^1\hfill & \text{from\hspace{0.17em}}110\hfill \\ {11}^2\hfill & \text{from\hspace{0.17em}}484\hfill \end{array}$
4. The LCM is the product of these numbers.
   
   $\begin{array}{lll}\text{LCM}\hfill & =\hfill & 2^2·3·5·{11}^2\hfill \\ \hfill & =\hfill & 4·3·5·121\hfill \\ \hfill & =\hfill & 7260\hfill \end{array}$