Question

What is the square root of $2601$ using the long division method?
A) $53$
B) $52$
C) $51$
D) $56$

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 $2601$ 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.
\[\overline{26}\overline{01}\]

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 $26$. So, $5$ is the largest number whose square is less than or equal to $26$. So, we will divide by $5$.
\[\begin{align}
  &\,\,\,\,\, 5 \\
 & 5\left| \!{\overline {\,
 \begin{align}
  & \overline{26}\overline{01} \\
 & \underline{25} \\
 & 1 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 3: Bring down the next pair. Here our next pair is $01$.
\[\begin{align}
  &\,\,\,\,\, 5 \\
 & 5\left| \!{\overline {\,
 \begin{align}
  & \overline{26}\overline{01}\\
 & \underline{25} \\
 & 101 \\
\end{align} \,}} \right. \\
\end{align}\]

Step 4: Now, double the value of the quotient and write it in 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 $10$. Now, if we take the next part of the divisor as $1$, we get the product $101 \times 1 = 101$.
So, we have,
\[\begin{align}
  &\,\,\,\,\, 51 \\
 & 5\left| \!{\overline {\,
 \begin{align}
  & \overline{26}\overline{01} \\
 & \underline{25} \\
 & 101 \\
\end{align}
 \,}} \right. \\
 & 101\left| \!{\overline {\,
 \begin{align}
  & 101 \\
 & \underline{101} \\
 & 0 \\
\end{align} \,}} \right. \\
\end{align}\]
Now, the quotient obtained in the long division process is the square root of the given number.
Hence, the square root of $2601$ is $51$. Thus, option (B) is the correct answer.

Note:
We must remember to place the decimal point once we arrive at the position of the decimal point in the dividend or original number, if it is a decimal number. The given question could also be solved with a smart guess. The original number is $2601$ . 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 is only one option $51$ with the leftmost digit as one or nine. Hence, option (B) is the correct answer.