Question

An auditorium has a rectangular array of chairs. There are exactly 14 boys seated in each row and exactly 10 girls seated in each column. If exactly 3 chairs empty, prove that there are at least 567 chairs in the auditorium.

Answer 1

lets say , there are 3 empty chairs and all in one row, that will imply that we have 14 colums and 13 rows .

that would mean that we have 182 chairs,
which is lesser that 567

Answer 2

That is just an example though. The question is asking for a solid proof

Answer 3

I believe I have a solution. Let the auditorium have m rows and n columns. Then there will be 14m boys, and 10n girls in the auditorium. There are 3 empty seats, and m*n total seats. This gives us the equation:\[mn=14m+10n+3\Longrightarrow mn-14m-10n=3\]\[(m-10)(n-14)-140=3\Longrightarrow (m-10)(n-14)=143\]Let:\[x=m-10,y=n-14\]The only integer solutions to the equation:\[xy=143\] with x and y positive are (143,1), (1,143), (11, 13), (13,11). This will generate pairs in m and n of:
(m,n) = (153, 15), (11, 157), (21, 27), (23,25).
Hence the total number off seats in the auditorium is at least 567 (which is the product of 21 and 27, the smallest product of the bunch).