Question

Find the least number which must be added to 6203 to obtain a perfect square. Also, find the square root of the number so obtained.

Answer 1

Hint: We first use the division of square root process to find the square root of the nearest square number of 6203. We use that number to find the closest square number greater than 6203. We find the difference and the square root of the number so obtained.

Complete step-by-step answer:
We first have to find and check if the number 6203 is a square number or not. For this we use the square root form.
We take 2 digits as a set from the right end and complete the division
\[\begin{align}
  & 78 \\
 & 7\left| \!{\overline {\,
 \begin{align}
  & \overline{62}\overline{03} \\
 & \underline{49} \\
\end{align} \,}} \right. \\
 & 148\left| \!{\overline {\,
 \begin{align}
  & 1303 \\
 & \underline{1148} \\
 & 155 \\
\end{align} \,}} \right. \\
\end{align}\]
From the division, we can see that the square root of the nearest square number of 6203 is 78.
So, $ {{78}^{2}}=6084 $
Therefore, the closest square number greater than 6203 will be the square of $ 78+1=79 $ .
We first find the square of 79.
So, $ {{79}^{2}}=6241 $ .
To find the least number which must be added to 6203 to obtain a perfect square will be the difference between 6241 and 6203.
The difference will be $ 6241-6203=38 $ .
The least number is 38 and the square root of the number so obtained is 79.

Note: We need to be careful about the question and check if it has asked about a lesser or greater square number. For a lesser square number we just have to subtract the remainder from the number to get the solution.