Question

If a,b,c are distinct prime integers such that ${{a}^{2}}{{b}^{3}}{{c}^{4}}=49392$, find the values of $a,b$ and $c$ .

Answer 1

Hint: Here in this question we have been asked to find the values of $a,b$ and $c$ which are distinct prime integers such that ${{a}^{2}}{{b}^{3}}{{c}^{4}}=49392$. For that we will prime factorize the given number 49392.

Complete step by step answer:
Now considering from the question we have been asked to find the values of $a,b$ and $c$ which are distinct prime integers such that ${{a}^{2}}{{b}^{3}}{{c}^{4}}=49392$.
For that we will prime factorize the given number 49392.
The prime factorization of 49392 can be done as follows:
Firstly we will use 2 the first prime number to factorize the given number 49392. By doing that we will have
$\begin{align}
  & 2\left| \!{\underline {\,
  49392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24696 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12348 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6174 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  3087 \,}} \right. \\
\end{align}$
As 3087 can’t be factorized using 2 we will use the next prime number 3. By doing this we will have
$\begin{align}
  & 2\left| \!{\underline {\,
  49392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24696 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12348 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6174 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3087 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1029 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  343 \,}} \right. \\
\end{align}$
As 343 can’t be factorized using 3 we will use the prime number 7 as 5 can’t be used. By doing this we will have
$\begin{align}
  & 2\left| \!{\underline {\,
  49392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24696 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12348 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6174 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3087 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1029 \,}} \right. \\
 & 7\left| \!{\underline {\,
  343 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence 49392 can be expressed as ${{2}^{4}}{{3}^{2}}{{7}^{3}}$ .

Hence we can conclude that the values of $a,b$ and $c$ which are distinct prime integers such that ${{a}^{2}}{{b}^{3}}{{c}^{4}}=49392$ are $3,7,2$ respectively.

Note: While answering questions of this type we should be sure with our calculations that we are going to perform in between the steps. Someone can unintentionally make a mistake and can consider the prime factorization result as ${{2}^{4}}{{3}^{3}}{{7}^{2}}$ which will lead us to end up having the conclusion as $a=7,b=3,c=2$ which is a wrong answer.