A teacher on attempting to arrange the students for mass drill in the form of a square found that 24

assume that there were x students ina row.

so the square will have

x (x) students.

 so total number of students =x(x)+24.....as 24 students get left out after this arrangement.

next

if the size of the square is increased by 1 student, then the  number of students in the new arangement=[(x+1)(x+1)]

so total number of students=[(x+1)(x+1)]-25.... as 25 students  were found short out after this arrangement.

since the total number of students is the same, so equating it in both cases we get,

x(x)+24=[(x+1)(x+1)]-25

so

2x=48

x=24

hence, the number of students=[24 x24]+24=576+24=600

or even in the other formation,

 number of students=[25 x 25]-25=600

so we see that the number of students=600