Question

Find the LCM of the following: ${{x}^{3}}{{y}^{2}}$, $xyz$?
(a) ${{x}^{3}}{{y}^{2}}z$
(b) ${{x}^{2}}{{y}^{2}}z$
(c) ${{x}^{3}}{{y}^{2}}{{z}^{2}}$
(d) None of these

Answer 1

Hint: We start solving the problem by recalling the definition of LCM of two or more given numbers as the smallest positive number that can be divisible by all those given numbers. We then divide the numbers with the common factors present in both the given numbers using the Ladder method. We then make the necessary calculations to get the required value of LCM.

Complete step-by-step solution
According to the problem, we need to find the LCM (Least common multiple) of the given terms: ${{x}^{3}}{{y}^{2}}$, $xyz$.
Let us recall the definition of LCM (Least Common Multiple) of given numbers. We know that the LCM (Least Common Multiple) of two or more given numbers is defined as the smallest positive number that can be divided by all of the given numbers.
Let us find the LCM of ${{x}^{3}}{{y}^{2}}$, $xyz$ by using Cake or Ladder method which is as shown below:
$\begin{align}
  & x\left| \!{\underline {\,
  {{x}^{3}}{{y}^{2}},xyz \,}} \right. \\
 & y\left| \!{\underline {\,
  {{x}^{2}}{{y}^{2}},yz \,}} \right. \\
 & \left| \!{\underline {\,
  {{x}^{2}}y,z \,}} \right. \\
\end{align}$.
Let us now multiply the remaining terms to get the LCM of ${{x}^{3}}{{y}^{2}}$, $xyz$.
So, the LCM of ${{x}^{3}}{{y}^{2}}$, $xyz$ is $x\times y\times {{x}^{2}}y\times z={{x}^{3}}{{y}^{2}}z$.
We have found that the LCM of ${{x}^{3}}{{y}^{2}}$, $xyz$ as ${{x}^{3}}{{y}^{2}}z$.
$\therefore$ The correct option for the given problem is (a).

Note: We can also solve this problem by finding GCF (Greatest Common Factor) first and then dividing the product of the two terms with the obtained GCF which will be equal to LCM. We should know that at least two terms in the given area to be divisible by any factor in order to divide the terms with that factor in the ladder method. Here we assumed that the given terms ${{x}^{3}}{{y}^{2}}$, $xyz$ are positive to proceed through the problem. Similarly, we can expect problems to find the GCF of the terms ${{x}^{3}}{{y}^{2}}$, $xyz$.