Question

What is the smallest number that should be subtracted from 506948 to get a perfect square? Find
the square root of the perfect square.
and show the prosses

Answer 1

$\green{\large\underline{\sf{Solution-}}}$

Given number is 506948

Now, we have to find a number that must be subtracted from 506948 to make it a perfect square.

So, we use Long Division method to evaluate the number

that must be subtracted from 506948 to make it a perfect square.

So, by using Long Division Method, we have

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:712 \:\:}}}\\ {\underline{\sf{7}}}& {\sf{\:\:506948 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \:49 \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{141}}}& {\sf{\:\: 169   \:\:}} \\{\sf{}}& \underline{\sf{\:\:141 \:  \:}} \\ {\underline{\sf{142}}}& {\sf{\: \: \: \: \: \: \: \:2848 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: 2844\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \: \: \: \: \: \: \: \:4\:\:}} \end{array}\end{gathered}$

So, it means 4 must be subtracted from 506948 to make it a perfect square.

So, required number = 506948 - 4 = 506944

Hence,

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:712 \:\:}}}\\ {\underline{\sf{7}}}& {\sf{\:\:506944 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \:49 \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{141}}}& {\sf{\:\: 169   \:\:}} \\{\sf{}}& \underline{\sf{\:\:141 \:  \:}} \\ {\underline{\sf{142}}}& {\sf{\: \: \: \: \: \: \: \:2844 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: 2844\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \: \: \: \: \: \: \: \:0\:\:}} \end{array}\end{gathered}$

$\rm \implies\:\boxed{\tt{ \sqrt{506944} \: = \: 712 \: }}$

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Additional Information :-

1. The number of digits in the square root of perfect square number consisting of 2n digits is n.

2. The number of digits in the square root of perfect square number consisting of 2n + 1 digits is n + 1.

3. The perfect square numbers can never ends with 2, 3, 7, 8 and odd number of zeroes.

4. Between squares of two consecutive natural numbers n and n + 1, 2n natural number lies.

Answer 2

Given number is 506948

Now, we have to find a number that must be subtracted from 506948 to make it a perfect square.

So, we use Long Division method to evaluate the number

that must be subtracted from 506948 to make it a perfect square.

So, by using Long Division Method, we have

$\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:712 \:\:}}}\\ {\underline{\sf{7}}}& {\sf{\:\:506948 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \:49 \: \: \: \: \: \: \:\:}} \\ {\underline{\sf{141}}}& {\sf{\:\: 169 \:\:}} \\{\sf{}}& \underline{\sf{\:\:141 \: \:}} \\ {\underline{\sf{142}}}& {\sf{\: \: \: \: \: \: \: \:2848 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: 2844\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \: \: \: \: \: \: \: \: \: \:4\:\:}} \end{array}\end{gathered}\end{gathered}$

So, it means 4 must be subtracted from 506948 to make it a perfect square.

So, required number = 506948 - 4 = 506944

Hence,

$\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\:712 \:\:}}}\\ {\underline{\sf{7}}}& {\sf{\:\:506944 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \:49 \: \: \: \: \: \: \:\:}} \\ {\underline{\sf{141}}}& {\sf{\:\: 169 \:\:}} \\{\sf{}}& \underline{\sf{\:\:141 \: \:}} \\ {\underline{\sf{142}}}& {\sf{\: \: \: \: \: \: \: \:2844 \:\:}} \\{\sf{}}& \underline{\sf{\:\: \: \: \: \: \: \: 2844\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \: \: \: \: \: \: \: \: \: \:0\:\:}} \end{array}\end{gathered}\end{gathered}$

$\rm \implies\:\boxed{\tt{ \sqrt{506944} \: = \: 712 \: }}⟹ 506944$

=712