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If p is prime number what is the lcm of $ p,{p^2},{p^3} $

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Last updated date: 24th Jul 2024
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Answer
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Hint: In the question we are told that the p is a prime number. A prime number is the number which has factors itself and 1.
In this case p has the factors p and 1. Follow the procedure of factorization and find lcm stepwise.

Complete step-by-step answer:
In this case $ p,{p^2},{p^3} $
P is a prime number. We can find the factors of each term as
 \[
  p = p \times 1 \\
  {p^2} = p \times p \times 1 \\
  {p^3} = p \times p \times p \times 1 \;
 \]
In this case we can observe the factors of the given terms and see if any common factors exist.
Taking common factors in all $ p,{p^2},{p^3} $ and continuing till the end.
The common in three and two of them will be multiplied only once and the remaining all factors will be multiplied to the result.
This will help us In getting the answers by applying the basic definition of lowest common multiple.
It is a multiple which can be divided by all the given numbers and will be the lowest of such categories to exist.
So, finally solving,
We get the final solution as $ {p^3} $ .
  $ {p^3} $ is the lcm of given numbers $ p,{p^2},{p^3} $ .
So, the correct answer is “ $ {p^3} $ ”.

Note: In the process of finding the lcm we must factorise the components. If in case hcf is given and we need to find the lcm product of two numbers = product of lcm and hcf.
Understand that hcf is a factor and lcm is multiple of given numbers. Almost every time lcm > hcf.