Question

# Find the square root of 2209.(a) 47(b) 23(c) 37(d) 53

Hint: In order to find the square roots of any number, the easiest way is to follow the synthetic division process. Firstly, we need to take the pair of the last two numbers and find the number whose square will form that pair. Here, we can take 09 as the pair, so the numbers that can form this number by squaring are 3 and 7. So, we will divide the number starting with 7. We can go on a hit and trial method in case the numbers that can form the pair by squaring is more than 1. So, we can also start division with 3 and check the result and so on. However, we will begin with 7 here and check whether the number is exactly divisible or not. If the number gets exactly divided and there remains no remainder, then the number is our required answer.

Here, we have to find the square root of 2209. We will solve this question using a synthetic division process. In order to find the divisor, we will take the pair of the last two numbers and find the number whose square will form that pair. Here, we can take 09 as the pair, so the numbers that can form this number by squaring are 3 and 7. So, we will take the divisor as 7 and check if it can exactly divide the given number or not. If it fails, we will try with 3. If the number is exactly divisible, then the quotient will be our required answer.

In synthetic division, we will divide as we divide normally. But every time we divide the number with any number, we will also add it to the current divisor. And on adding, the result obtained will be our new divisor. This process goes on till the time we have a very small remainder which is not further divisible or when we have no remainder left. Hence, the quotient will be the required answer, only if we have no remainder left.
Step 1:
\begin{matrix} 4 \\ 4 \\ \end{matrix}\left| \begin{align} & 2209 \\ & 16 \\ \end{align} \right|47
Step 2:
\begin{align} & \underline{\begin{matrix} 4 \\ 4 \\ \end{matrix}\left| \begin{align} & 2209 \\ & 16 \\ \end{align} \right|4} \\ & 8\left| \text{ }609 \right| \\ \end{align}
Step 3:
\begin{align} & \text{ }\underline{\begin{matrix} 4 \\ 4 \\ \end{matrix}\left| \begin{align} & 2209 \\ & 16 \\ \end{align} \right|4}7 \\ & \underline{\begin{matrix} 87 \\ 7 \\ \end{matrix}\left| \begin{align} & \times 609 \\ & \text{ }609 \\ \end{align} \right|\text{ }} \\ & 94\left| \text{ }\times \times \times \right| \\ & \text{ } \\ \end{align}
Since, the remainder is ‘zero’, so the square root of 2209 will be 47.

So, the correct answer is “Option A”.

Note: Students often make mistakes in solving the above division. They need to be aware of the actual division process. There is an alternate method to solve these questions.
There are some steps which can be used to find the square root of a four – digit number: (i) We can firstly take the pair of the last two numbers and find the number whose square will form that pair. Here, we can take 09 as the pair, so the numbers that can form this number by squaring are 3 and 7. Hence, the unit digit number of the square root either will be 3 or 7. (ii) Then, we can consider the first two pairs of the number. Here, the first two pairs of number are 22, which comes between the square of 4 and 5, i.e.,
$\left( {{4}^{2}}=16 \right)<\left( {{5}^{2}}=25 \right)$
. Hence, from the given options, option (c) and option (d) are cancelled out. So, the remaining option is option (a) and option (d). Since the number is smaller than the square of 5 which is 25, so we can try to find the square of option (a). On squaring the option (a), we will find the answer. Besides, students can easily eliminate the option using the trick mentioned above in (i) and (ii). This is an easy way that can save time and easily be solved.