hugogva wrote:
A certain marching band has fewer than 50 members. During performances, the members march in different formations. When they march in Formation A, each row has 5 members, except the last row, which has 3 members. When they march in Formation B, each row has 6 members, except the last row, which has 5 members. If they march in Formation C, each row has 7 members, except the last row. How many band members are in the last row in Formation C?
A) 1
B) 2
C) 3
D) 4
E) 6
Let the number of members in the band = n
When they march in Formation A, each row has 5 members, except the last row, which has 3 members.\(n = 5x_1 + 3\)
\(x_1\) → Number of rows that have 5 members
When they march in Formation B, each row has 6 members, except the last row, which has 5 members.\(n = 6x_2 + 5\)
\(x_2\) → Number of rows that have 6 members
Combining both of the information, we can combine
\(n = \text{LCM}(6,5)x + \text{first common term}\)
\(n = 5x_1 + 3\) ⇒ 3, 8, 13, 18, 23, ....
\(n = 6x_2 + 5\) ⇒ 5, 11, 17, 23, ....
Therefore
\(n = \text{LCM}(6,5)x + 23\)
As the number of members is fewer than 50, the number of members in the marching band = 23
Question:
If they march in Formation C, each row has 7 members, except the last row. How many band members are in the last row in Formation C?Inference: The question requires us to find the remainder when the number of members in the marching band, i.e. 23, is divisible by 7.
\(23 = 7 * 3 + 2\)
Remainder = \(2\)
Option B