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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?
Let's start by understanding what each formation tells us about the total number of band members.
In Formation A: Members march with 5 people per row, except the last row has 3 members. This means if we divide the total number of members by 5, we get a remainder of 3.
In Formation B: Members march with 6 people per row, except the last row has 5 members. This means if we divide the total number of members by 6, we get a remainder of 5.
In Formation C: Members march with 7 people per row, except the last row has some number we need to find. This means we need to find what remainder we get when dividing the total by 7.
We also know there are fewer than 50 members total.
Process Skill: TRANSLATE - Converting the formation descriptions into mathematical remainder conditions
Now let's express these conditions mathematically:
If we call the total number of members N, then:
Let's work through this systematically. We need a number that gives remainder 3 when divided by 5 and remainder 5 when divided by 6.
Numbers that give remainder 3 when divided by 5: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, ...
Now let's check which of these also gives remainder 5 when divided by 6:
So N = 23 is our answer! Let's verify:
Process Skill: APPLY CONSTRAINTS - Systematically checking all possibilities under 50
Now that we know there are 23 band members total, let's see what happens in Formation C where each row has 7 members except the last row.
\(23 \div 7 = 3 \text{ remainder } 2\)
This means there are 3 complete rows of 7 members each (\(3 \times 7 = 21\) members), and the last row has 2 members (\(23 - 21 = 2\)).
The last row in Formation C has 2 members.
The answer is B) 2.
Students often misunderstand what "each row has X members, except the last row" means. They might think the last row always has fewer members than X, when actually it could have any number from 1 to X-1. This leads to incorrect remainder interpretations.
Many students fail to translate the formation descriptions into remainder conditions (\(\mathrm{N} \equiv 3 \pmod{5}\), \(\mathrm{N} \equiv 5 \pmod{6}\)). Instead, they might try to set up complex equations or use trial-and-error without a systematic approach.
Students sometimes focus only on the formation conditions and forget to use the upper limit constraint, which is crucial for finding the unique solution efficiently.
When checking each candidate number, students frequently make division errors or incorrectly calculate remainders, especially when working with numbers like \(23 \div 6 = 3 \text{ remainder } 5\).
Students may start checking numbers that satisfy the first condition (remainder 3 when divided by 5) but stop too early or skip numbers, missing the correct total of 23 members.
Even when finding N = 23, students might incorrectly verify by making calculation mistakes when checking \(23 \div 5\) and \(23 \div 6\), leading them to doubt their correct answer.
After correctly finding that there are 23 total members, students might make errors in the final step: \(23 \div 7 = 3 \text{ remainder } 2\). They might miscalculate this division or confuse which number represents the last row.
Students might correctly calculate \(23 \div 7 = 3 \text{ remainder } 2\) but then select 3 (the number of complete rows) instead of 2 (the number of members in the last row), misreading what the question asks for.