Question

What least number must be added to 700 to make the sum a perfect square? (Use the Long division method).

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 700 to make the sum a perfect square?
This can find by using a method of inverse operation of squaring.
Now, find square of 700 using a division method i.e.,
\[\begin{align}
 & {26 \\ 2}\left| \!{\overline {\,
 \begin{align}
&\overline{7} \ \overline{00} \\
 & \underline{4} \\
 & 3 \\
\end{align} \,}} \right. \\
 & 46\left| \!{\overline {\,
 \begin{align}
  & 3\overline{00} \\
 & \underline{276} \\
 & 24 \\
\end{align} \,}} \right. \\
\end{align}\]
 So the square root of 700 is 26 and the remainder is 24.
As we know, the square number of 26 is 676 i.e., $${26^2} = 676$$.
We need to add to 700 to make it a perfect square. But 676 is less than 700.
So let us take square of successor number i.e., $${27^2} = 729$$
Therefore, the difference of 729 and 700 is
$$ \Rightarrow \,\,729 - 700$$
$$\therefore \,\,29$$
Thus, when 29 is added to 700 we get 729.
Hence, 729 is a perfect square.
So, the correct answer is “729”.

Note: The above solved method is known as finding the square root is 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.