Question

Find the least number which must be subtracted from 4215 to make it a perfect square?

Answer 1

Hint:
Here, we will try to find the square root of 4215 by division method. If we get the remainder as 0, then 4215 will be a perfect square but if the remainder is any number other than 0, then it will not be a perfect square. Subtracting the remainder from 4215 will give us the required number which is a perfect square. Hence, the remainder obtained will be the least number which must be subtracted from 4215 to make it a perfect square.

Complete step by step solution:
According to the question, first of all, we will try to find the square root of the given number 4215, by using the Division method.
 \[\begin{array}{*{20}{l}}{}{64}\\6{\left| \!{\overline {\,
 \begin{array}{l}\overline {42} \overline {15} \\ - 36\end{array} \,}} \right. }\\{124}{\left| \!{\overline {\,
 \begin{array}{l}615\\\underline { - 496} \\119\end{array} \,}} \right. }\end{array}\]
In this method, we start pairing the numbers by taking a bar starting from the unit’s place. We take the largest possible number whose square will be less than or equal to the leftmost pair.
Now, we double the value of the quotient and assume it as a ten’s place digit and for the one’s place we have to take a digit such that multiplying the total number by that digit, we get a number less than or equal to the carried down number. Since, the remainder is 119 and not 0 in this question, hence, 119 is the least number which should be subtracted from 4215 to make it a perfect square and hence, get the remainder 0.

Therefore, 119 is the required least number.

Note:
We can check whether our answer is correct or not by first of all, subtracting the remainder i.e. 119 from the given number.
Hence, we get, \[\left( {4215 - 119} \right) = 4096\]
Now, we will find the square root of this number by either the division method above or by doing its prime factorization.
Hence, prime factorization of 4096 is:
\[\begin{array}{*{20}{l}}
  2\| \underline {4096} \\

  2\| \underline {2048} \\

  2\| \underline {1024} \\

  2\| \underline {512} \\

  2\| \underline {256} \\

  2\| \underline {128} \\

  2\| \underline {64} \\

  2\| \underline {32} \\

  2\| \underline {16} \\

  2\| \underline 8 \\

  2\| \underline 4 \\

  2\| \underline 2 \\

  {}\| \underline 1
\end{array}\]

Hence, 4096 can be written as:
\[4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Now, since, we are required to find the square root,
We will take only one prime number out of a pair of the same prime numbers.
\[ \Rightarrow \sqrt {4096} = 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow \sqrt {4096} = 64\]
Hence, the square root of 4096 is 64.
Therefore, it is a perfect square.
Hence, verified