Question

For the following number, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
1458

Answer 1

Hint:Here first we will factorize the given number by using the method of prime factorization and then find the number which is not in pair. The number so obtained would be the smallest number which should be multiplied in order to make the given number a perfect square.Then we will factorize the number obtained by multiplying the given number with the number so obtained and then find its square root.

Complete step-by-step answer:
The given number is 1458
Factorizing the given number by using the method of prime factorization we get:-
$$\,\,\,\,\,\,
  \begin{array}{|l}
    \llap{2~~~~} 1458 \\ \hline
    \llap{3~~~~} 729 \\ \hline
    \llap{3~~~~} 243 \\ \hline
    \llap{3~~~~} 81 \\ \hline
    \llap{3~~~~} 27 \\ \hline
    \llap{3~~~~} 9 \\ \hline
    \llap{3~~~~} 3 \\ \hline
                 1
  \end{array}
\,\,\,\,\,\,
$$
\[1498 = 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\]
Now as we can see that each of the number is making pair with other except for 2
And since in order to make the given number a perfect square we need to make the pair of each of the factors.
Hence we need to make a pair of 2 in order to make it a perfect square.
Hence 2 is the smallest number which should be multiplied in order to make the 1498 a perfect square.
 Now multiplying 1498 by 2 we get:-
 \[1498 \times 2 = 2996\]
Factorizing the number so obtained we get:-
$$\,\,\,\,\,\,
  \begin{array}{|l}
   \llap{2~~~~} 2996 \\ \hline
    \llap{2~~~~} 1458 \\ \hline
    \llap{3~~~~} 729 \\ \hline
    \llap{3~~~~} 243 \\ \hline
    \llap{3~~~~} 81 \\ \hline
    \llap{3~~~~} 27 \\ \hline
    \llap{3~~~~} 9 \\ \hline
    \llap{3~~~~} 3 \\ \hline
                 1
  \end{array}
\,\,\,\,\,\,
$$
\[2996 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\]
Taking the square of 2996 we get:-
 \[
  \sqrt {2996} = \sqrt {2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3} \\
  \sqrt {2996} = 2 \times 3 \times 3 \times 3 \\
 \]
Multiplying and simplifying it further we get:-
\[
  \sqrt {2996} = 2 \times 3 \times 3 \times 3 \\
  \sqrt {2996} = 6 \times 3 \times 3 \\
  \sqrt {2996} = 18 \times 3 \\
  \sqrt {2996} = 54 \\
 \]
Hence the square root of 2996 is 54.

Note:Students should note that prime factorization is the method of factorizing a number in the multiples of prime numbers.Also, the square root of a perfect square is always a whole number.