If there are B boys and G girls in a club, can the girls be divided equally among 6 teams...

Understanding the Question

The question asks: Can the girls be divided equally among 6 teams with no girls left over?

In other words, we need to determine whether \(\mathrm{G}\) (the number of girls) is divisible by 6. Since \(\mathrm{6 = 2 \times 3}\), this means \(\mathrm{G}\) must be divisible by both 2 and 3.

For this yes/no question to be sufficient, we need to definitively answer either "Yes, \(\mathrm{G}\) is divisible by 6" or "No, \(\mathrm{G}\) is not divisible by 6" for all possible cases.

Given information:

• \(\mathrm{B}\) = number of boys in the club
• \(\mathrm{G}\) = number of girls in the club
• We need to determine: Is \(\mathrm{G}\) divisible by 6?

Analyzing Statement 1

Statement 1 tells us: "If there were 4 fewer girls, then the number of girls would be twice the number of boys."

This translates to: \(\mathrm{G - 4 = 2B}\), which means \(\mathrm{G = 2B + 4}\)

Let's think about what this tells us about divisibility:

• Since \(\mathrm{G = 2B + 4 = 2(B + 2)}\), we know \(\mathrm{G}\) is always even (divisible by 2) ✓
• But what about divisibility by 3?

Let's test different scenarios based on \(\mathrm{B}\)'s remainder when divided by 3:

• If \(\mathrm{B \equiv 0 \pmod{3}}\), like \(\mathrm{B = 3}\): \(\mathrm{G = 2(3) + 4 = 10}\) → \(\mathrm{10 \div 3 = 3}\) remainder 1 → not divisible by 3
• If \(\mathrm{B \equiv 1 \pmod{3}}\), like \(\mathrm{B = 4}\): \(\mathrm{G = 2(4) + 4 = 12}\) → \(\mathrm{12 \div 3 = 4}\) remainder 0 → divisible by 3
• If \(\mathrm{B \equiv 2 \pmod{3}}\), like \(\mathrm{B = 5}\): \(\mathrm{G = 2(5) + 4 = 14}\) → \(\mathrm{14 \div 3 = 4}\) remainder 2 → not divisible by 3

Since different values of \(\mathrm{B}\) lead to different answers about whether \(\mathrm{G}\) is divisible by 6, Statement 1 alone cannot tell us definitively whether the girls can be divided equally among 6 teams.

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: "If the number of boys were 2 less than twice the actual number of boys, then the boys could be divided equally among 6 teams with no boys left over."

This means \(\mathrm{2B - 2}\) is divisible by 6.

What can we deduce from this?

• \(\mathrm{2B - 2 = 2(B - 1)}\) must be divisible by 6
• For \(\mathrm{2(B - 1)}\) to be divisible by 6, the factor \(\mathrm{(B - 1)}\) must be divisible by 3
• Therefore, \(\mathrm{B - 1 \equiv 0 \pmod{3}}\), which means \(\mathrm{B \equiv 1 \pmod{3}}\)

But here's the crucial point: This tells us something about \(\mathrm{B}\)'s divisibility properties, but nothing about \(\mathrm{G}\)! Without any relationship between \(\mathrm{B}\) and \(\mathrm{G}\), we cannot determine whether \(\mathrm{G}\) is divisible by 6.

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B (and confirms D is already eliminated).

Combining Both Statements

Now let's use both pieces of information together:

• From Statement 1: \(\mathrm{G = 2B + 4}\)
• From Statement 2: \(\mathrm{B \equiv 1 \pmod{3}}\) (\(\mathrm{B}\) leaves remainder 1 when divided by 3)

Let's trace through the divisibility:

• Since \(\mathrm{B \equiv 1 \pmod{3}}\), we know \(\mathrm{B}\) leaves remainder 1 when divided by 3
• This means \(\mathrm{2B}\) leaves remainder 2 when divided by 3 (since \(\mathrm{2 \times 1 = 2}\))
• Therefore, \(\mathrm{2B + 4}\) leaves remainder \(\mathrm{2 + 4 = 6}\) when divided by 3
• But \(\mathrm{6 \equiv 0 \pmod{3}}\), so \(\mathrm{G}\) is divisible by 3 ✓

We already established from Statement 1 that \(\mathrm{G = 2(B + 2)}\) is always even (divisible by 2) ✓

Since \(\mathrm{G}\) is divisible by both 2 and 3, it must be divisible by 6. We can definitively answer "Yes" - the girls can be divided equally among 6 teams.

The combined statements are SUFFICIENT.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Both statements together are sufficient to determine that the girls can be divided equally among 6 teams, but neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."