Question

If \[A,B\] and \[C\] are three numbers such that L.C.M. of \[A\] and \[B\] is \[B\] and the L.C.M. of \[B\] and \[C\] is \[C\] then the L.C.M. of \[A,B\] and \[C\] is
A.\[A\]
B.\[B\]
C.\[C\]
D.\[\dfrac{{A + B + C}}{3}\]

Answer 1

Hint: Here, we will find the least common multiple for three numbers. First, we will find the relation between the three numbers is given by the least common multiple of two numbers. The Least Common Multiple (L.C.M) is defined as the smallest number that is divisible by both the numbers.

Complete step-by-step answer:
We are given that the L.C.M. of \[A\] and \[B\] is \[B\].
So, we have \[B\] is a multiple of \[A\].
We are given that the L.C.M. of \[B\] and \[C\] is \[C\].
So, we have \[C\] is a multiple of \[B\].
Now, we will find the L.C.M. of \[A,B\] and \[C\].
Since \[B\] is a multiple of \[A\] and \[C\] is a multiple of \[B\], then by transitive property, we have \[C\] as a multiple of \[A\].
Thus the L.C.M. of \[A,B\] and \[C\] is \[C\] which is the multiple of both \[A\] and \[B.\]
Thus, option(C) is the correct answer.

Note: If two numbers are given and one number is a multiple of another number, then the largest number would be the LCM of two numbers. Similarly, if \[n\] numbers are given and all the numbers are the multiples of a number, then the largest number from all the numbers would be the L.C.M. of all the numbers. Transitive Property states that “If \[a\] is related to \[b\] by the property and \[b\] is related to \[c\] by the same property, then \[a\] is related to \[c\] by the same property.” i.e., if \[a = b\] and \[b = c\], then \[a = c\].