Question

Find the LCM of the following: ${{x}^{2}}y+x{{y}^{2}},{{x}^{2}}+xy$
A. $xy\left( x+y \right)$
B. $xy\left( x-y \right)$
C. ${}^{2}\left( x+y \right)$
D. None of these

Answer 1

Hint: We will first start by factoring the terms ${{x}^{2}}y+x{{y}^{2}},{{x}^{2}}+xy$ by taking the variable common in both the terms. Then we will use the fact that LCM or least common multiple of two numbers is the least number which is divisible by both the numbers. So, we take the highest power of the common factors and the other factors of both the number for this.

Complete step-by-step answer:
Now, we have to find the LCM of ${{x}^{2}}y+x{{y}^{2}},{{x}^{2}}+xy$.
So, using factorization we can write,
$\begin{align}
  & {{x}^{2}}y+x{{y}^{2}}=xy\left( x+y \right) \\
 & =x\times y\times \left( x+y \right) \\
 & {{x}^{2}}+xy=x\left( x+y \right) \\
 & =x\times \left( x+y \right) \\
\end{align}$
Now, we know that the LCM of two numbers is the least number which is completely divisible by both numbers. So, for this we take the highest power of common factors of both the terms and other terms to find the LCM.
So, LCM of ${{x}^{2}}y+x{{y}^{2}},{{x}^{2}}+xy$ is $xy\left( x+y \right)$.
Hence, the correct option is (B).

Note: It is important to note that we have taken $\left( x+y \right)$ only once not twice as we have to find the least number which is divisible by both ${{x}^{2}}y+x{{y}^{2}}\ and\ {{x}^{2}}+xy$ and therefore, $xy\left( x+y \right)$ is least number which is divisible by both ${{x}^{2}}y+x{{y}^{2}}\ and\ {{x}^{2}}+xy$. The number $xy{{\left( x+y \right)}^{2}}$ is also divisible by both ${{x}^{2}}y+x{{y}^{2}}\ and\ {{x}^{2}}+xy$ but it is not the least number. Hence, that would be wrong and the correct answer is $xy\left( x+y \right)$.