Question

A high school marching band can be arranged in a rectangular formation with exactly three boys in each row and exactly five girls in each column. There are several sizes of marching band for which this is possible. What is the sum of all such possible sizes?

Answer 1

312

Answer 2

Let the number of rows and columns be r and c respectively. Although both r and c can be any quantity, we know that each row must contain exactly 3 boys and each column exactly 5 girls. Therefore, we get 3r+5c=rc. Rearranging, rc-3r-5c=0. Then

r(c-3)-5c=0
r(c-3)-5c+15=15
r(c-3)-5(c-3)=15
(r-5)(c-3)=15

So now we can substitute two integers that multiply to 15 to find (r-5) and (c-3).

r-5=1, r=6 and c-3=15, c=18
r-5=3, r=8 and c-3=5, c=8
r-5=5, r=10 and c-3=3, c=6
r-5=15, r=20 and c-3=1, c=4

With the values of r and c, multiply them to find the possible sizes and add to find the answer.

6 x 18 = 104
8 x 8 = 64
10 x 6 = 60
20 x 4 = 80

Hence, the sum of all possible sizes is 104+64+60+80=312