Question

$5929$ students are sitting in an auditorium in such a manner that there are as many students in a row as there are rows in the auditorium. How many rows are there in the auditorium?

Answer 1

Hint:
We are given that $5929$ students are sitting in an auditorium in such a manner that there are as many students in a row as there are rows in the auditorium. Now, consider the number of students in a row as $a$. After that, calculate the number of students in $a$ rows and so keep it equal to $5929$. Try it, you will get the answer.

Complete step by step solution:
In the problem we are given that $5929$ students are sitting in an auditorium in such a manner that there are as many students in a row as there are rows in the auditorium.
Here, let us consider the number of students in a row$=a$.
Now let us find the number of students in $a$ row, we get,
Number of students in $a$ row$=a\times a={{a}^{2}}$
Here, Number of students in $a$ row$=$ total number of students in auditorium
Now, it is given that the total number of students in the auditorium$=5929$.
From above we can see that, ${{a}^{2}}=5929$
Now taking square root on both sides we get,
$a=\sqrt{5929}$
Now, factors of $5929$$=7\times 7\times 11\times 11$
Simplifying we get,
factors of $5929$$={{7}^{2}}\times {{11}^{2}}$ ………. (By prime)
Now we know that, $a=\sqrt{5929}$ , so substituting factors of $5929$$={{7}^{2}}\times {{11}^{2}}$ in $a=\sqrt{5929}$ we get,
$a=\sqrt{{{7}^{2}}\times {{11}^{2}}}$
So, we get,
$a=7\times 11$
Simplifying in simple form we get,
$a=77$
Therefore, the number of rows in the auditorium is $77$.

Additional information:
Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. Prime factorization is defined as a way of finding the prime factors of a number i.e. the prime numbers which can be multiplied together to get the original number. The prime numbers when multiplied by any natural numbers or whole numbers (but not $0$), gives composite numbers. So basically prime factorization is performed on the composite numbers to factorize them and find the prime factors.

Note:
Here we have given the total number of students. We have assumed the number of students in row as $a$. After that, a prime factorization method is used. Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers.