Answer

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**Hint:**First we will assume that the number of rows is \[x\] and then the number of columns is \[x\]. Now, we know that the total number of students is computed by finding the product of number of rows and number of columns and then simplify using the long division method to find the remainder.

**Complete step-by-step answer:**We are given that there are 500 children in a school and the number of rows is equal to the number of columns.

Let us assume that the number of rows is \[x\].

Thus, the number of columns is \[x\].

Now, we know that the total number of students is computed by finding the product of number of rows and number of columns. So, we have

\[

\Rightarrow 500 = x \times x \\

\Rightarrow 500 = {x^2} \\

\Rightarrow {x^2} = 500 \\

\]

Taking the square root on both sides of the above equation, we get

\[ \Rightarrow x = \sqrt {500} \]

Since the above number is a larger number with 3 digits, we will find the square of 500 using the long division method.

We know that the square root of a given number using the division method is calculated by grouping the digits in pairs starting from the right to left, Then we will now take the largest number whose square is equal to or just less than the first pair. Then subtract the product of the obtained divisor and the quotient from the first pair and bring down two the new pair. We will continue this process until the remainder is zero.

First, we will separate the digits by taking bars from right to left once in two digits of the given number 500.

\[\overline 5 {\text{ }}\overline {00} \]

Now we will find the square root of the above pairs using the long division method.

Here, the remainder is 16.

We need to find the least number that must be subtracted from 500 so as to get a perfect square.

Thus, we will subtract 16 from the 500 to get a perfect square.

**Therefore, 16 students will be left out in this arrangement.**

**Note:**We can also solve this problem using the prime factorization, we get

\[ \Rightarrow 500 = 2 \times 2 \times 5 \times 5 \times 5\]

Taking square root on both sides of the above equation and simplifying, we get

\[

\Rightarrow \sqrt {500} = \sqrt {2 \times 2 \times 5 \times 5 \times 5} \\

\Rightarrow \sqrt {500} = 2 \times 5 \times \sqrt 5 \\

\Rightarrow \sqrt {500} = 10 \times 2.03 \\

\Rightarrow \sqrt {500} = 22.03 \\

\Rightarrow \sqrt {500} \approx 22 \\

\]

We have that in each row or each column maximum 22 students can be arranged, so the maximum students in P.T. is \[{\left( {22} \right)^2}\].

Subtracting maximum students from the total number of students, we get

\[

\Rightarrow 500 - {22^2} \\

\Rightarrow 500 - 484 \\

\Rightarrow 16 \\

\]

Thus, 16 students will be left out in this arrangement.

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