Question

Find the LCM of the following:
\[{{a}^{2}}bc,{{b}^{2}}ac,{{c}^{2}}ab\]
\[\left( \text{a} \right)\text{ }{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
\[\left( \text{b} \right)\text{ }{{a}^{2}}{{b}^{3}}{{c}^{2}}\]
\[\left( \text{c} \right)\text{ }{{a}^{3}}{{b}^{2}}{{c}^{2}}\]
(d) None of these

Answer 1

Hint: To solve the above question, we will first find out what an LCM is. We will first write the squared terms as the product of the variables in all the three numbers given in the question. After writing this, we will select the term which is common to all the terms and then we will multiply it with the remaining terms which are not common in them. The product of these variables will be equal to the LCM of three numbers.

Complete step-by-step answer:
Before solving the question, we must know what an LCM is. The full form of LCM is the least common multiple or the lowest common multiple. The LCM of two or more integers is the smallest positive integer that is divisible by all the numbers which we want to take LCM of. Now, first we will write \[{{a}^{2}}bc\] as the product of the factors. Thus, we have,
\[{{a}^{2}}bc=a\times a\times b\times c.....\left( i \right)\]
Similarly, we write the other two terms as the product of factors. Thus, we can say that,
\[{{b}^{2}}ac=b\times b\times a\times c\]
\[{{c}^{2}}ab=c\times c\times a\times b\]
Now, we have,
\[{{a}^{2}}bc=a\times b\times c\times a=\left( a\times b\times c \right)\times a\]
\[{{b}^{2}}ac=a\times b\times c\times b=\left( a\times b\times c \right)\times b\]
\[{{c}^{2}}ab=a\times b\times c\times c=\left( a\times b\times c \right)\times c\]
Now, here we can see that the term \[\left( a\times b\times c \right)\] is common to all three terms. Now, we will multiply this term \[\left( a\times b\times c \right)\] to the remaining terms (a, b and c) to get the LCM of the three numbers given in the question. Thus, we have the following.
\[\text{LCM of }{{a}^{2}}bc,{{b}^{2}}ac,{{c}^{2}}ab=\left( a\times b\times c \right)\times a\times b\times c\]
\[\Rightarrow \text{LCM of }{{a}^{2}}bc,{{b}^{2}}ac,{{c}^{2}}ab=\left( a\times a \right)\left( b\times b \right)\left( c\times c \right)\]
\[\Rightarrow \text{LCM of }{{a}^{2}}bc,{{b}^{2}}ac,{{c}^{2}}ab={{a}^{2}}{{b}^{2}}{{c}^{2}}\]
Hence, option (a) is the right answer

Note: While solving the question, we have assumed that the variables a, b and c are not zero and not negative. This means that they are positive integers because the LCM of the negative integers and zero does not exist. For us to find the LCM of the given numbers they should be positive integers. Once we write the factors of the three terms, we can directly write the LCM as the product of factors raised to power of the maximum number of times that particular factor is appearing in the factorisation of three terms. Here, we have all factors a, b, c appearing 2 times in factorisation of three terms. So, we can write as
\[\Rightarrow \text{LCM of }{{a}^{2}}bc,{{b}^{2}}ac,{{c}^{2}}ab={{a}^{2}}{{b}^{2}}{{c}^{2}}\]