Question

Write the HCF and LCM of the smallest odd composite number and the smallest odd prime number. If an odd number $p$ divides ${{q}^{2}}$, then will it divide ${{q}^{3}}$ also? Explain.

Answer 1

Hint: To solve this question, we should know the concept of HCF, that is the highest common factor and LCM, that is least common multiple. Also, we should know that composite numbers are those which have more than 2 factors and prime numbers are those that have only 2 factors.

Complete step-by-step solution -
In order to solve this question, we should know that composite numbers are those that have factors other than 1 and the number itself, whereas prime numbers are those that have 1 and the number itself as the only 2 factors. So, we can say that the smallest odd composite number is 9 because $9=1\times 3\times 3$ or we can say that the factors of 9 = 1, 3, 9. The smallest odd prime number is 3 because factors of 3 = 1, 3. Now, we have to find the HCF and LCM of 3 and 9. So, we will write,
$\begin{align}
  & 9=3\times 3 \\
 & 3=1\times 3 \\
\end{align}$
Now, we know that HCF is the largest possible number that can divide both the given numbers. And from the factorisation of 3 and 9, we can say that 3 is the HCF of 3 and 9.
We also know that LCM is the smallest possible number that is a multiple of them. So, from the factorisation of 3 and 9, we can say that 9 is the LCM of 3 and 9. Hence, we can say that the HCF and LCM of the smallest odd composite number and the smallest odd prime number are 3 and 9 respectively.
Now, we have been asked if an odd number $p$ divides ${{q}^{2}}$, then will it divide ${{q}^{3}}$ also. For that, we will consider the given condition that is, $p$ divides ${{q}^{2}}$. So, we can write it as, ${{q}^{2}}=pm$. Now we will consider ${{q}^{3}}$. We can write ${{q}^{3}}$ as ${{q}^{2}}\times q$. We know that, ${{q}^{2}}=pm$, so we can write ${{q}^{3}}$ as, ${{q}^{3}}=pm\times q$. This implies that $p$ is the factor of ${{q}^{3}}$.
Hence, we can say that if $p$ divides ${{q}^{2}}$, then will it definitely divide ${{q}^{3}}$ also.

Note: We can solve the second part of the question by considering the given condition, that is $p$ divides ${{q}^{2}}$. And we know that if $p$ divides ${{q}^{2}}$, then it will definitely divide $q$, and from this we can say that $p$ will divide ${{q}^{3}}$ also.