Question

What is the square root of $10201$ using the long division method?
A) $101$
B) $99$
C) $103$
D) $98$

Answer 1

Hint: In the given question, we need to find the square root of the given number using the long division method. Square root of a number is a number that when multiplied with itself yields the original number. There are various steps involved in the process of calculating the square root of a number using the long division process as discussed in the solution.

Complete step by step solution:
We will find the square root of the number $10201$ using the long division method.
Steps to find square root using long division:
Step 1: Write down the number whose square root is to be found and place a bar over the pairs of two numbers, starting from the left side.
\[\overline{1}\overline{02}.\overline{01}\]

Step 2: For the divisor, take the largest number whose square is less than or equal to the first pair of numbers. Here, the first pair is $1$. So, $1$ is the largest number whose square is less than or equal to $1$. So, we will divide by $1$.
\[\begin{align}
  &\,\,\,\,\,\, 1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{1}\overline{02}\overline{01} \\
 & \underline{1} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 3: Bring down the next pair. Here our next pair is $02$.
\[\begin{align}
  &\,\,\,\,\,\, 1 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{1}\overline{02}\overline{01} \\
 & \underline{1} \\
 & 002 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 4: Now, double the value of the quotient and write it in the divisor. For the second digit, we need to write such a number which when multiplied by the new number obtained gives us less than or equal to the dividend.
Here the quotient is $1$. So, the first part of the divisor will be $2$. Now, if we take the next part of the divisor as $0$, we get the product $20 \times 0 = 0$, which is less than $2$.
So, we have,
\[\begin{align}
  &\,\,\,\,\,\, 10 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{1}\overline{02}\overline{01} \\
 & \underline{1} \\
 & 002 \\
\end{align}
 \,}} \right. \\
 & 20\left| \!{\overline {\,
 \begin{align}
  & 2 \\
 & \underline{0} \\
 & 2 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 5: Now, taking the next pair down. Now, we get the first part of the divisor by doubling the quotient, that is $10 \times 2 = 20$.
Now, if we take the next part of the divisor as $1$, we get the product $201 \times 1 = 201$.
So, we get,
\[\begin{align}
  &\,\,\,\,\,\, 101 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{1}\overline{02}\overline{01} \\
 & \underline{1} \\
 & 0 \\
\end{align}
 \,}} \right. \\
 & 20\left| \!{\overline {\,
 \begin{align}
  & 2 \\
 & \underline{0} \\
 & 201 \\
\end{align} \,}} \right. \\
 & 201\left| \!{\overline {\,
 \begin{align}
  & 201 \\
 & \underline{201} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
Now, the quotient obtained in the long division process is the square root of the given number.
Hence, the square root of $10201$ is $101$. Thus, option (A) is the correct answer.

Note:
The given question could also be solved with a smart guess. The original number is $10201$ that means that the leftmost digit of the square root of the number must be either one or nine, since we know that numbers ending in $1$ or $9$ have the leftmost digit of square as $1$.
So, there are only two options satisfying the condition: $99$ and $101$.
Now, we know that a square of $100$ is equal to $10000$.
We can observe that the given number $10201$ is greater than $10000$. SO, the square root must be greater than a hundred. Hence, option (A) is the correct answer.