A certain marching band has fewer than 50 members. During performances, the members march in different formations. When they march...

1. Translate the problem requirements: We need to find the number of band members (less than 50) based on how they arrange in different formations, where each formation leaves different remainders when divided by 5, 6, and 7.
2. Set up remainder conditions from the formation descriptions: Convert the formation descriptions into mathematical conditions about what remainder the total gives when divided by each formation size.
3. Find the total number satisfying all conditions: Systematically check numbers less than 50 that satisfy both remainder conditions from formations A and B.
4. Apply the total to Formation C: Once we know the total number of members, determine how many are in the last row when arranged in groups of 7.

Execution of Strategic Approach

1. Translate the problem requirements

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

2. Set up remainder conditions from the formation descriptions

Now let's express these conditions mathematically:

• When the total number of members is divided by 5, the remainder is 3
• When the total number of members is divided by 6, the remainder is 5
• The total is less than 50

If we call the total number of members N, then:

• \(\mathrm{N} \equiv 3 \pmod{5}\) - meaning N leaves remainder 3 when divided by 5
• \(\mathrm{N} \equiv 5 \pmod{6}\) - meaning N leaves remainder 5 when divided by 6
• \(\mathrm{N} < 50\)

3. Find the total number satisfying all conditions

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:

• \(3 \div 6 = 0 \text{ remainder } 3\) ✗
• \(8 \div 6 = 1 \text{ remainder } 2\) ✗
• \(13 \div 6 = 2 \text{ remainder } 1\) ✗
• \(18 \div 6 = 3 \text{ remainder } 0\) ✗
• \(23 \div 6 = 3 \text{ remainder } 5\) ✓
• \(28 \div 6 = 4 \text{ remainder } 4\) ✗
• \(33 \div 6 = 5 \text{ remainder } 3\) ✗
• \(38 \div 6 = 6 \text{ remainder } 2\) ✗
• \(43 \div 6 = 7 \text{ remainder } 1\) ✗
• \(48 \div 6 = 8 \text{ remainder } 0\) ✗

So N = 23 is our answer! Let's verify:

• \(23 \div 5 = 4 \text{ remainder } 3\) ✓ (Formation A: 4 complete rows of 5, plus 3 in last row)
• \(23 \div 6 = 3 \text{ remainder } 5\) ✓ (Formation B: 3 complete rows of 6, plus 5 in last row)
• \(23 < 50\) ✓

Process Skill: APPLY CONSTRAINTS - Systematically checking all possibilities under 50

4. Apply the total to Formation C

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 (\(23 - 21 = 2\)).

Final Answer

The last row in Formation C has 2 members.

The answer is B) 2.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting formation descriptions

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.

2. Not recognizing the modular arithmetic pattern

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.

3. Overlooking the constraint "fewer than 50 members"

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.

Errors while executing the approach

1. Arithmetic errors in remainder calculations

When checking each candidate number, students frequently make division errors or incorrectly calculate remainders, especially when working with numbers like \(23 \div 6 = 3 \text{ remainder } 5\).

2. Incomplete systematic checking

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.

3. Verification errors

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.

Errors while selecting the answer

1. Confusing the final calculation

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.

2. Selecting the wrong component of the answer

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.