Question

Find the consecutive perfect squares between which the following numbers lie: 56789

Answer 1

Hint: First, find the square root of the number by long division method, i.e. we take the largest number as the divisor whose square is less than or equal to the number on the extreme left. The number on the extreme left is the dividend. Then we divide and write the quotient and so on until no divided left and the remainder is less than the divisor. The square of the quotient and the square of the number next to the quotient are the desired perfect squares.

Complete step-by-step solution:
Given:- The number is 56789
Now, find the square root of 56789 by the long division method.
Separate the digits by taking bars from right to left after two digits.
$\overline 5 \,\overline {67} \,\overline {89} $
Now, multiply a number by itself such that the product must be less than or equal to 5. The condition will be met by “2” and the remainder will be 1. Then,
$\begin{gathered}
  \left. 2 \right)\overline 5 \,\overline {67} \,\overline {89} \left( 2 \right. \\
  \,\,\,\,\,\underline 4 \\
  \,\,\,\,\,1 \\
\end{gathered} $
Now, bring down 67. The new dividend is 167. Now double the quotient to get a new divisor. Also, concatenate it with a suitable digit such that the product of the new divisor and the digit is less than or equal to the new dividend.
\[\begin{gathered}
  \left. {\,\,\,2} \right)\overline 5 \,\overline {67} \,\overline {89} \left( {23} \right. \\
  \,\,\,\,\,\,\,\,\underline 4 \\
  \left. {43} \right)167 \\
  \,\,\,\,\,\,\,\,\underline {129} \\
  \,\,\,\,\,\,\,\,\,\,38 \\
\end{gathered} \]
Now, bring down 89. The new dividend is 3889. Now again double the quotient to get a new divisor. Also, concatenate it with a suitable digit such that the product of the new divisor and the digit is less than or equal to the new dividend.
\[\begin{gathered}
  \,\,\,\,\,\left. 2 \right)\overline 5 \,\overline {67} \,\overline {89} \left( {238} \right. \\
  \,\,\,\,\,\,\,\,\,\,\underline 4 \\
  \,\,\left. {43} \right)167 \\
  \,\,\,\,\,\,\,\,\,\,\underline {129} \\
  \left. {438} \right)3889 \\
  \,\,\,\,\,\,\,\,\,\,\underline {3504} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,385 \\
\end{gathered} \]
Thus, the square root of the number 56789 is greater than 238 but less than 239. So,
$ \Rightarrow 238 < \sqrt {56789} < 239$
Square the expression,
$\therefore {\left( {238} \right)^2} < 56789 < {\left( {239} \right)^2}$

Hence, the consecutive perfect squares between which 56789 lie is 238 and 239.

Note: Steps of Long Division Method for Finding Square Roots:
First, group the digits in pairs. Start with the digit in the units place.
Take the largest number whose square is equal to or just less than the first period. Then, take the number as the divisor and also as the quotient.
Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.
Now, the new divisor is obtained by taking two times the quotient and concatenating with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
Repeat the steps (2), (3), and (4) till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.