Question

The LCM of \[{{a}^{2}}-{{b}^{2}}\] and \[\left( a-b \right)\] is:
A. \[{{a}^{2}}-{{b}^{2}}\]
B. \[\left( a-b \right)\]
C. \[\left( b-a \right)\]
D. \[{{b}^{2}}-{{a}^{2}}\]

Answer 1

Hint: We know that the LCM of the given expression is the smallest expansion that is divisible by each of the given expressions. Now to find the LCM of the two given expressions, first of all we will express each of the expressions as the product of its factors and then we will find the product of each factor with the highest power which occurs in the given expression. This product is equal to LCM of the two given expressions.

Complete step-by-step answer:
We have been given the two expressions: \[{{a}^{2}}-{{b}^{2}}\] and \[\left( a-b \right)\].
Now we will factorize each of the expressions.
We already know that \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\] and \[\left( a-b \right)=\left( a-b \right)\].
After factoring each of the expressions, find the LCM now we will find the product of each factor with the highest power which occurs in the given expression. This product is equal to the LCM of the given expression.
So the highest power of \[\left( a+b \right)\] is 1 and the highest power of another factor \[\left( a-b \right)\] is also 1.
Hence, LCM,
\[\begin{align}
  & ={{\left( a+b \right)}^{1}}\times {{\left( a-b \right)}^{1}} \\
 & =\left( a+b \right)\left( a-b \right) \\
 & ={{a}^{2}}-{{b}^{2}} \\
\end{align}\]
Hence, LCM of \[\left( {{a}^{2}}-{{b}^{2}} \right)\] and \[\left( a-b \right)\] is equal to \[{{a}^{2}}-{{b}^{2}}\].
Therefore, the correct answer of the above question is option A.

Note: Be careful of the sign while factorizing the given expression. Also remember that LCM has full form least common multiple and the forms of LCM of two numbers a and b is as follows:
LCM (a, b) \[=\dfrac{\left| a.b \right|}{\gcd (a,b)}\] where gcd is the greatest common divisor.