Question

If a, b, and c are distinct positive prime integer such that $ {{a}^{2}}{{b}^{3}}{{c}^{4}}=49392 $ , then find value of a, b, and c.

Answer 1

Hint:
To solve the given equation we will use the prime factorization method and do the prime factorization of 49392 and then we will compare the power of coefficients obtained after factorization with the power of a, b and c.

Complete step by step answer:
We will start solving the above question by using the prime factorization method and from the question, we know that $ {{a}^{2}}{{b}^{3}}{{c}^{4}}=49392 $ where a, b, and c are the distinct prime number.
So, the prime factorization of 49392 is equal to:
 $ \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. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align} $
So, we can say that the prime factorization of 49392 = $ {{2}^{4}}\times {{3}^{2}}\times {{7}^{3}} $
Now, from question we know that $ {{a}^{2}}{{b}^{3}}{{c}^{4}}=49392 $ , where a, b and c all are prime number.
So, we can also write $ {{2}^{4}}\times {{3}^{2}}\times {{7}^{3}} $ = $ {{a}^{2}}{{b}^{3}}{{c}^{4}}=49392 $
So, after comparing the power of 2, 3, and 7 with a, b, and c we can say that c = 2, because both c and 2 have the same power, that is 4. Similarly, b = 7 because both b and 7 have power 3 and also a = 3 because both a and 3 have power 2.
Hence, a = 3, b = 7 and c = 2 and 2, 3 and 7 all are also prime.
This is our required solution.

Note:
Students are required to note that they do not make any calculation mistakes while doing prime factorization and while comparing the powers. Also, note that when we do the prime factorization, we always the number with the prime number not by composite, and also any number can be expressed as a multiple of a prime number.