Question

Find the least number which must be added to 8400 to obtain a perfect square. Find the perfect square and its square root.

Answer 1

Hint: We perform the square root operation on the non-square value of 8400. From the operation, we get the nearest square number less than 8400. We just add 1 to that root value to get the nearest square number greater than 8400. We perform subtraction to find the least number which must be added to 8400.

Complete step by step solution:
We need to find the root of the given number 8400 although it’s a non-square number. We form the process in a remainder process. We use that remainder to find the nearest square number greater than 8400.
First, we take dual digits as a single input from the left side of 8400. So, $\overline{84}\overline{00}$.
Now we try to form the square of a fixed number to go near each input.
$9\times 2\overset{9}{\overline{\left){\begin{align}
  & \overline{84}\overline{00} \\
 & \underline{81} \\
 & 0300 \\
\end{align}}\right.}}$
Now we need to put such a digit after 9 so that the multiplication doesn’t cross 300.
$181\overset{91}{\overline{\left){\begin{align}
  & \overline{84}\overline{00} \\
 & \underline{81} \\
 & 0300 \\
 & \underline{0181} \\
 & 0119 \\
\end{align}}\right.}}$
In this case to get to near 84 we chose 9 and we took twice of that value as 18 for the next part.
In the remainder we got 300. We have to place such a value after 9 which is also being placed after 18 and multiplied. We don't cross 300. Such value is 1 as if we place 2, we get the multiplied value $2\times 182=364$ which is greater than 300.
So, the nearest square number less than 8400 will be the square of 91.
Square value of 91 will be ${{91}^{2}}=91\times 91=8281$.
We look out for the next square number which will be square of 92.
Square value of 91 will be ${{92}^{2}}=92\times 92=8464$.
This is greater than 8400. To reach 8464, we need to add $8464-8400=64$.
Therefore, we need to add 64. The perfect square is 8464 and the root value is 92.

Note: The process of adding 1 is to get to the least number which must be added to 8400 or to get the greater square number. But to get the lesser square number or the number to subtract we just need to find the remainder. We need to subtract that number only.