Question

If two positive integers a and b are expressible in the form of \[a=p{{q}^{2}}\] and \[b={{p}^{3}}q\]; p,q being prime numbers, then LCM (a, b) is

\[A.\,pq\]
\[B.\,{{p}^{3}}{{q}^{3}}\]
\[C.\,{{p}^{3}}{{q}^{2}}\]
\[D.\,{{p}^{2}}{{q}^{2}}\]

Answer 1

Hint: We will make use of the LCM using the repeated division method to find the value of the LCM of the given terms a, b which are expressed in the form of the prime numbers p, q, as \[a=p{{q}^{2}}\] and \[b={{p}^{3}}q\]. We will divide the given numbers a and b by the common factors until no further division is possible.

Complete step by step answer:
From the data, we have the data as follows.
The terms p, q are the prime numbers. The terms a, b are the positive integers expressed in terms of the prime numbers p, q as follows.
\[a=p{{q}^{2}}\]and\[b={{p}^{3}}q\]
We are supposed to find the least common multiple of these numbers.
There are three methods to solve this problem, that is, to find the LCM. We will consider the one method, named as LCM using the repeated division.
In this type of method, we will successively divide both the numbers by the common factors, until the remainder doesn’t get divided further.
Let us start with the division process.
\[\begin{align}
  & p\left| \!{\underline {\,
  p{{q}^{2}},\,{{p}^{3}}q \,}} \right. \\
 & \,\,\,\left| \!{\underline {\,
  {{q}^{2}},\,{{p}^{2}}q \,}} \right. \\
\end{align}\]
In the above operation, we have divided the numbers by p, as both the terms contain ‘p’ as the common term.
The remainders obtained are \[{{q}^{2}},\,{{p}^{2}}q\].
As these remainders are further divisible, we will continue the division process.
\[\begin{align}
  & p\left| \!{\underline {\,
  p{{q}^{2}},\,{{p}^{3}}q \,}} \right. \\
 & q\,\left| \!{\underline {\,
  {{q}^{2}},\,{{p}^{2}}q \,}} \right. \\
 & \,\,\,\,\left| \!{\underline {\,
  q,\,{{p}^{2}} \,}} \right. \\
\end{align}\]
In the above operation, we have divided the numbers by q, as both the terms contain ‘q’ as the common term.
The remainders obtained are \[q,\,{{p}^{2}}\].
As these remainders are not further divisible, we will stop the division process.
Now we will compute the LCM.
The LCM is the product of the numbers that we have used to divide and the remainders.
So, we have,
\[\begin{align}
  & \downarrow p\left| \!{\underline {\,
  p{{q}^{2}},\,{{p}^{3}}q \,}} \right. \\
 & \downarrow q\,\left| \!{\underline {\,
  {{q}^{2}},\,{{p}^{2}}q \,}} \right. \\
 & \,\,\,\,\,\,\,\,\left| \!{\underline {\,
  q,\,{{p}^{2}} \,}} \right. \\
 & \,\,\,\,\,\,\,\,\,\to \to \\
\end{align}\]
The arrow marks represent the terms to be multiplied to obtain the LCM of the numbers.
So, we get,
\[\begin{align}
  & LCM(a,b)=p\times q\times q\times {{p}^{2}} \\
 & \Rightarrow LCM(a,b)={{p}^{3}}\times {{q}^{2}} \\
\end{align}\]
As, the value of the LCM (a, b) of the numbers \[a=p{{q}^{2}}\] and \[b={{p}^{3}}q\] is \[{{p}^{3}}{{q}^{2}}\].

So, the correct answer is “Option C”.

Note:
There are mainly three methods to solve this type of problem, that is, to find the LCM of given two numbers. They are LCM using the prime factorization – to divide the given numbers by the prime factors, LCM using the repeated division – to divide the given numbers by the common factors, until no further division is possible.