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# If two positive integers $'a'$ and $'b'$ are expressible in the form of $a = p{q^2}$ and $b = {p^3}q$; $'p'$ and $'q'$ being prime numbers, find the L.C.M. of $'a'$ and $'b'$.

Last updated date: 07th Sep 2024
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Hint:
Here, we have to find the least common multiple for the two positive integers. First we will find the factor form for the two positive integers. The two positive integers are expressed in the form of prime numbers. We will find the L.C.M for two positive integers using the factor form. Least Common Multiple is defined as the smallest integer that is a multiple of both of the integers.

Complete step by step solution:
We are given with two positive integers $a$ and $b$.
These two integers are expressed in the form of $a = p{q^2}$ and $b = {p^3}q$ where $p$ and $q$ are prime numbers.
Now, we will represent the integers in the form of factors.
Factor form of two integers are
$a = p \times q \times q$
$b=p\times p\times p\times q$
Now, we will find the L.C.M of two integers
$L.C.M.\left( a,b \right)=p\times q\times q\times p\times p$
$\Rightarrow L.C.M.\left( {a,b} \right) = {p^3}{q^2}$

Therefore, the L.C.M of two integers is ${{p}^{3}}{{q}^{2}}$.

Note:
In order to find the LCM, we need to remember some of its properties.
1) The properties of least common multiple include:
2) The L.C.M. of any two or more numbers cannot be less than any one of the numbers.
3) If a number is a factor of another number, then the number which is greater would be the L.C.M. of two numbers.
4) The L.C.M of two numbers where one number is prime to another number, then their L.C.M. would be either the product or the number which is greater.
5) The L.C.M of two consecutive numbers is their product.