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

Let there be ${}^{\prime }{a}^{\prime }$ rows in the auditorium
Since the number of students in a row is same as the number of rows in the auditorium.
$\therefore$ Number of students in a row $=a$
$\Rightarrow$ Number of students in ${}^{\prime }{a}^{\prime }$ rows $=a\times a=a^2$
It is given that the total number of students in the auditorium $=5929$.
$\therefore a^2=5929$
$\Rightarrow a=\sqrt{5929}$
$7\mid 5929\overline{7\mid 847}\overline{11\mid 121}\overline{\mid 11}$
$\Rightarrow a=\sqrt{(7\times 7)\times (11\times 11)}$
[By prime factorisation]
$\Rightarrow a=7\times 11$ [Taking one factor in each pair]
$\Rightarrow a=77$
Hence, there are $77$ rows in the auditorium.