Question

Find the LCM of the following: \[3\left( {a - 1} \right),2{\left( {a - 1} \right)^2},\left( {{a^2} - 1} \right)\]
\[
  {\text{A}}{\text{. 6}}{\left( {{\text{a - 1}}} \right)^{\text{2}}}\left( {{\text{a + 1}}} \right) \\
  {\text{B}}{\text{. 6}}{\left( {{\text{a + 1}}} \right)^{\text{2}}}\left( {{\text{a + 1}}} \right) \\
  {\text{C}}{\text{. 6}}{\left( {{\text{a - 1}}} \right)^{\text{2}}}\left( {{\text{a - 1}}} \right) \\
  {\text{D}}{\text{. None of the Above}} \\
 \]

Answer 1

Hint: LCM is elaborated as Least Common Multiple, which is the lowest common factor among the integers. To find the LCM of the given numbers, which is the lowest common number and is divisible by all numbers for which we are finding the LCM, the method includes basic factorization of the numbers to find factors that are multiplied together to form a number. If all the given numbers are \[0\] for which LCM is being calculated, their LCM will also be \[0\]. First, find the factors of the number that are the number when multiplied together gives the original number and then, multiply the common factors among the numbers to get the Least Common Factor (LCM) of the numbers.

Complete step by step answer:
To find the lowest common multiple of the numbers, let’s find the factors of the given numbers for which we have to find the LCM of each number.
Factors are the integers, which, when multiplied, results in the original number only. First, find the factors of numbers \[3\left( {a - 1} \right),2{\left( {a - 1} \right)^2},\left( {{a^2} - 1} \right)\]
\[3\left( {a - 1} \right) = 3 \times \left( {a - 1} \right)\]
\[2{\left( {a - 1} \right)^2} = 2 \times \left( {a - 1} \right) \times \left( {a - 1} \right)\]
\[\left( {{a^2} - 1} \right) = \left( {{a^2} - {1^2}} \right) = \left( {a + 1} \right) \times \left( {a - 1} \right)\] \[\left[ {\because \left( {{x^2} - {y^2}} \right) = \left( {x + y} \right)\left( {x - y} \right)} \right]\]
The unique common multiple all the numbers number is \[\left( {a - 1} \right)\]
Now multiply the common multiple with the other factors to calculate the LCM:
\[
  LCM = \left( {a - 1} \right) \times \left[ {2 \times 3 \times \left( {a - 1} \right) \times \left( {a + 1} \right)} \right] \\
   = 6\left( {a + 1} \right){\left( {a - 1} \right)^{^2}} \\
 \]
Hence, the LCM of the numbers \[3\left( {a - 1} \right),2{\left( {a - 1} \right)^2},\left( {{a^2} - 1} \right)\] is \[6\left( {a + 1} \right){\left( {a - 1} \right)^{^2}}\], thus Option ‘A’ is the correct answer.

Additional Information: The function ${\left( {a - 1} \right)^2}$ can be written as \[\left( {a - 1} \right)\left( {a + 1} \right)\].

Note: In general, LCM is the common multiple of the numbers, which will divide all the numbers participating in the solution completely without leaving any remainder. Methods of finding LCM are Prime factorization, repeated division, using multiples.