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.
Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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
That is just an example though. The question is asking for a solid proof
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).