Question

What would be added to 7912 to make the sum a perfect square?

Answer 1

Hint: Here in this question, we have to find what we have to add such a given number can form a perfect square. This can be solved by, first we need to find the square root and their reminder of a given number using a division method and later by the difference of the square of its quotient number and the square of their successor number we get the required solution.

Complete step-by-step answer:
When a number or integer (not a fraction) is multiplied by itself, the resultant is called a ‘Square Number’.
Consider the given question
What would be added to 7912 to make the sum a perfect square?
This can be found by using a method of inverse operation of squaring.
Now, find square of 7912 using a division method i.e.,
\[\begin{align}
 & {88 \\ 8}\left| \!{\overline {\,
 \begin{align}
&\overline{79} \ \overline{12} \\
 & \underline{64} \\
 & 15 \\
\end{align} \,}} \right. \\
 & 168\left| \!{\overline {\,
 \begin{align}
  & 15\overline{12} \\
 & \underline{1344} \\
 & 168 \\
\end{align} \,}} \right. \\
\end{align}\]
 So the square root of 7912 is 88 and remainder is 168.
As we know, the square number of 88 is 7744 i.e., \[{88^2} = 7744\].
We need to add to 7912 to make it a perfect square. But 7744 is less than 7912.
So let us take square of successor number i.e., \[{89^2} = 7921\]
Therefore, the difference of 7921 and 7721 is
\[ \Rightarrow \,\,7921 - 7912\]
\[\therefore \,\,9\]
Thus when 9 is added to 7912 we get 7921.
Hence, 7921 is a perfect square.
So, the correct answer is “9”.

Note: The above solved method is known as finding the square root by the inverse operation of squaring. The inverse (opposite) operation of addition is subtraction and the inverse operation of multiplication is division. And we should know the squares of numbers at least 1 to 10 which helps to solve any problems based on square number or square root.