Question

2304 students are sitting in a lecture room in such a manner that there are as many

students in a row as there are rows in a lecture room. How many students are there

in each row of a lecture room?

Answer 1

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

Given that,

2304 students are sitting in a lecture room in such a manner that there are as many students in a row as there are rows in a lecture room.

So, Let assume that

Number of students in each row = x

So,

Number of rows in lecture room = x

Given that,

Total number of students in lecture room = 2304

$\rm :\longmapsto\:x \times x = 2304$

$\rm :\longmapsto\: {x}^{2} = 2304$

$\rm\implies \:x = \sqrt{2304}$

Now, to find the square root of 2304 we use the Method of Prime factorization.

So,

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:2304 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:1152\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:576\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:288 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:144 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:72 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:36 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:18 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:6\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:2 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

$\rm\implies \:2304 = {2}^{8} \times {3}^{2}$

$\rm\implies \: \sqrt{2304} = \sqrt{ {2}^{8} \times {3}^{2} }$

$\rm\implies \: \sqrt{2304} = {2}^{4} \times {3}^{2}$

$\rm\implies \: \sqrt{2304} = 16 \times 3$

$\rm\implies \: \sqrt{2304} = 48$

So, it means Number of students in each row = 48

Answer 2

Given: 2304 students are sitting in a lecture room in such a manner that there are as many  students in a row as there are rows in a lecture room.

To find: Number of students in each row.

Solution:

Observe that from the question,

2304 students are sitting in a lecture room in such a manner that there are as many students in a row as there are rows in a lecture room.

Assume that , Number of students in each row $= x$

Therefore, Number of rows in lecture room = x

Understand that,  

Total number of students in lecture room $= 2304$

$\[\begin{array}{l} \Rightarrow x \times x = 2304\\ \Rightarrow {x^2} = 2304\\ \Rightarrow x = \sqrt {2304} \end{array}\]$

Find the square root of 2304.

$\[\begin{array}{l} \Rightarrow 2304 = {2^8} \times {3^2}\\ \Rightarrow \sqrt {2304} = \sqrt {{2^8} \times {3^2}} \\ \Rightarrow \sqrt {2304} = {2^4} \times 3\\ \Rightarrow \sqrt {2304} = 16 \times 3\\ \Rightarrow \sqrt {2304} = 48\end{array}\]$

Hence, Number of students in each row$= 48$.