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?
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To find the sum of all possible sizes of the marching band, we need to determine the possible dimensions (length and width) of the rectangular formation.
Let's start by finding the factors of the number of boys per row (3) and the number of girls per column (5).
The factors of 3 are 1 and 3.
The factors of 5 are 1 and 5.
Since there are three boys per row and five girls per column, the width must be a factor of 3 (number of boys per row) and the length must be a factor of 5 (number of girls per column).
Now, let's consider the possible combinations of factors:
Width = 1, Length = 1:
In this case, we have a 1x1 formation, which means there is only one person in the band (1 boy and 1 girl). The sum is 1.
Width = 3, Length = 1:
We cannot have a width of 3 (number of boys per row) because the minimum possible length of 5 would not be satisfied. Hence, this combination is not possible.
Width = 1, Length = 5:
In this case, we have a 1x5 formation, which means there are 5 people in the band (3 boys and 5 girls). The sum is 5.
Width = 3, Length = 5:
In this case, we have a 3x5 formation, which means there are 15 people in the band (9 boys and 15 girls). The sum is 15.
Therefore, the sum of all possible sizes of the marching band is 1 + 5 + 15 = 21.