Question

By what number should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number.
 $ 16562 $

Answer 1

Hint: To see whether the given number is a perfect square or not. We will make its prime factors or we write a given number as a product of prime factors. If prime factors are not in pair then it will not be a perfect square number and then we will divide by those numbers which are not in pair to make the given number perfect square.

Complete step by step solution:
Given number is $ 16562 $
We first make prime factors of the given number
 $ \begin{array}{*{20}{c}}
  2&\hline & {16562} \\
\hline
  7&\hline & {8281} \\
\hline
  7&\hline & {1183} \\
\hline
  {13}&\hline & {169} \\
\hline
  {13}&\hline & {13} \\
\hline
  {}&\hline & 1
\end{array} $
Hence, prime factors of $ 16562 $ are $ 2 \times 7 \times 7 \times 13 \times 13 $
In prime factor we see that the number $ 7\,\,and\,\,13 $ are in pair but number $ 2 $ hasn’t its pair.
Therefore, we can say that the number $ 16562 $ is not a perfect square number.
To make $ 16562 $ a perfect square number we will divide it by the number $ 2 $ as it hasn’t its pair in prime factors. So we will remove it by dividing the given number by $ 2 $ .
 $
   \Rightarrow \dfrac{{2 \times 7 \times 7 \times 13 \times 13}}{2} \\
   \Rightarrow 7 \times 7 \times 13 \times 13 \;
  $
After division all numbers are in pairs. Therefore, the number obtained on their product will be a perfect square number.
Therefore, a new perfect square number after dividing by $ 2 $ is $ 8281 $ . Which is a perfect square of a number $ 91 $ .

Note: If a number is not a perfect square then there are different ways to make it a perfect square number. We can make a number perfect square by multiplying it with some number, or by dividing it by some number or by subtracting some number from it or by adding some number to it. But if any method is mentioned in question then we have to proceed calculation according to that method which is asked in the problem.