Question

The LCM of \[{a^k},\,{a^{k + 3}},{a^{k + 5}}\] where \[k \in N\]is
A) \[{a^{k + 9}}\]
B) \[{a^k}\]
C) \[{a^{k + 6}}\]
D) \[{a^{k + 5}}\]

Answer 1

Hint:
In this question, we will use the prime factorization method to find out LCM. In step 1: We will express each number as a product of prime factors. In step 2: The product of highest powers of all the prime factors.

Complete step by step solution:
Here, we are given with \[{a^k},\,{a^{k + 3}},{a^{k + 5}}\] and we have to find the L.C.M.
So, we will proceed this question step by step:
So, Step1-
Now, we will use the Prime Factorization method to solve \[{a^k},\,{a^{k + 3}}\] and \[{a^{k + 5}}\]
Firstly, we are solving \[{a^k}\] in terms of prime factorization.
\[{a^k}\] = \[a{\rm{ }} \times a{\rm{ }} \times {\rm{ }}a{\rm{ }} \ldots \ldots .{\rm{ }}k\] times
Secondly, we are solving \[{a^{k + 3}}\] in terms of prime factorization.
\[{a^{k + 3}}\]= \[a{\rm{ }} \times a{\rm{ }} \times {\rm{ }}a{\rm{ }} \ldots \ldots .{\rm{ }}k{\rm{ }} + {\rm{ }}3\] times
And at last, we are solving \[{a^k}\] in terms of prime factorization.
\[{a^{k + 5}}\]= \[a{\rm{ }} \times a{\rm{ }} \times {\rm{ }}a{\rm{ }} \ldots \ldots .{\rm{ }}k{\rm{ }} + {\rm{ 5}}\] times
Thus, Step2-
Now, we will select the highest power of each prime factor
And hence, LCM = (\[{a^k},\,{a^{k + 3}},\,{a^{k + 5}}\]) = \[{a^{k + 5}}\] (Ans.)

Hence, the correct option is (D) \[{a^{k + 5}}\]

Additional Information:
As the number consists of two types of factors and multiple that are H.C.F that is the highest common factor and another one is L.C.M that is least common multiple. The question can be of L.C.M or H.C.F or to find a product of two numbers which is equal to (H.C.F of two numbers \[ \times \] L.C.M of two numbers).

Note:
LCM (Lowest Common Factor) is a method to find the smallest common multiple between any two or more numbers. You have two method to find LCM are as follows: 1) Prime Factorization method 2) Division method (we divide the number with least Prime number until remainder is a prime number or 1)