Question

Find the LCM of the following numbers: 24, 36, 45, and 60.

Answer 1

Hint: We will factorize all the given numbers in prime factors. Then we will make a list of the prime factors that are common to all the four given numbers and uncommon to the given four numbers. We will find the least common factor of the given four numbers by multiplying the highest powers of the common factors and the uncommon factors and get the desired answer.

Complete step by step solution:
Let us factorize the number 24. We can factorize 24 as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Now, we will write the prime factorization for the number 36 in the following manner,
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Next, we will factorize 45 into its prime factors as follows,
\[\begin{align}
  & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
We have to factorize the number 60 into its factors. We will write the prime factors of 60 in the following manner,
$\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Now, we will make a list of the factors that are common to all the four numbers 24, 36, 45 and 60. From the factorizations, we can see that 3 is the only common factor for all the four numbers. The highest power of 3 in the factorizations of all four numbers is ${{3}^{2}}$. Now, the uncommon factors of the four numbers are 2 and 5. The highest powers in the factorizations of the four numbers of these factors are ${{2}^{3}}$ and ${{5}^{1}}$. Now, to obtain the least common multiple, we will multiply the highest powers of the common and uncommon factors of the four numbers. Therefore, we get the following,
$\begin{align}
  & \text{LCM = }{{\text{3}}^{2}}\times {{2}^{3}}\times {{5}^{1}} \\
 & =9\times 8\times 5 \\
 & =360
\end{align}$
Hence, the LCM of 24, 36, 45, and 60 is 360.

Note: It is necessary that we understand the concept of the least common multiple and the highest common factor. The LCM is the number, which can be divided by the given four numbers, whereas HCF is the number that divides all the four given numbers. It is useful to write the prime factorizations of the numbers explicitly so that we can avoid making any minor errors and obtain the correct answer.