Question

Which least number must be subtracted to 899 to make a perfect square? (Use a long division method).
(a) 55
(b) 56
(c) 57
(d) 58

Answer 1

Hint: At first take the number 899, then start pairing up the digits from the right side. Then take the largest number whose square is less than or equal to the first left out digit or pair of digits and then subtract from it while taking the number as quotient. After this find a new divisor by adding the quotient with divisor. After subtraction, take the next pair until all pairs are exhausted and the remainder is the number which should be subtracted from the original number to get the answer. Divide 784 by 2, then remainder by first taking (2 \[\times \] 2 = 4) and then 48 as divisor and continue in the same manner.

Complete step-by-step answer:
In the question we have to find the least number which should be the least number which should be subtracted from 899 to make it a perfect square.
First we write the steps of doing long division method for finding square roots:
Step 1: First group the digits in pairs starting with the digit in units place. Each pair and the remaining digit (if any) is called a period.
Step 2: Think of the largest number whose square is equal to or just less than the first period. Taking this number as a divisor and also the quotient.
Step 3: 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 makes a new dividend.
Step 4: Now, the new divisor is obtained by taking two times the quotient and canceling the suitable digit which is also t when as the next digit of quotient, chosen in such a way that the product of new divisor and the digits is equal or just less than the new dividend.
Step 5: Repeat steps (2), (3), (4) till all the periods have been taken up. Now the quotient so obtained is the required square root of the given number.
Let take an example of find square root of 784,
\[2\overline{\left){\begin{align}
  & \overline{7}\overline{84} \\
 & \underline{-4} \\
 & 48\left| \!{\underline {\,
  \begin{align}
  & 384 \\
 & -384 \\
\end{align} \,}} \right. \\
 & 0 \\
\end{align}}\right.}\left| \!{\underline {\,
  28 \,}} \right. \]
Here square root is 28.
If the remainder comes, that should be subtracted to make the number perfect square and its quotient the number’s square root.
So the given number is 899.
\[2\overline{\left){\begin{align}
  & 899 \\
 & \underline{-4} \\
 & 49\left| \!{\underline {\,
  \begin{align}
  & 499 \\
 & -441 \\
\end{align} \,}} \right. \\
 & 58 \\
\end{align}}\right.}\left| \!{\underline {\,
  29 \,}} \right. \]
So ‘58’ should be subtracted so that the number becomes perfect square root 29.

So, the correct answer is “Option d”.

Note: Generally students miss out certain simple steps like pairing numbers from right hand side, also sometimes do calculations mistakes while doing subtraction. Also they can check whether their answer is right or wrong by subtracting the given number with the square of the quotient, if the result is equal to remainder then the answer is okay.