When all the boxes in a warehouse were arranged in stacks of 8, there were 4 boxes left over. If...

Understanding the Question

We need to find the exact number of boxes in a warehouse.

Given Information

• When boxes are arranged in stacks of 8, there are 4 boxes left over
• Total number of boxes is between 80 and 120 (exclusive)

What We Need to Determine

If we call the number of boxes N, then:

• N = 8k + 4 for some integer k (meaning N leaves remainder 4 when divided by 8)
• \(\mathrm{80 < N < 120}\)

This means N could be: 84, 92, 100, 108, or 116.

For a statement to be sufficient, it must narrow these 5 possibilities down to exactly ONE value.

Analyzing Statement 1

Statement 1 tells us: If boxes were arranged in stacks of 9, there would be no boxes left over.

This means N is divisible by 9.

Key Insight

We need a number that:

• Leaves remainder 4 when divided by 8
• Is divisible by 9 (leaves remainder 0)

Here's the crucial insight: Since 8 and 9 share no common factors (they're coprime), these two conditions create a very restrictive pattern. Numbers satisfying both conditions only occur every 72 numbers (\(\mathrm{8 × 9 = 72}\)).

In our narrow range of 80-120 (a span of just 40 numbers), there can be at most ONE such number.

Let's verify: Among our candidates \(\mathrm{\{84, 92, 100, 108, 116\}}\), which is divisible by 9?

• Quick check: \(\mathrm{108 ÷ 9 = 12}\) ✓

Only 108 works!

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

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

Statement 2 tells us: If boxes were arranged in stacks of 12, there would be no boxes left over.

This means N is divisible by 12.

Strategic Analysis

We need a number that:

• Leaves remainder 4 when divided by 8
• Is divisible by 12

Notice that \(\mathrm{12 = 4 × 3}\). Since \(\mathrm{N = 8k + 4}\) can be rewritten as \(\mathrm{N = 4(2k + 1)}\), we know N is always divisible by 4. So the real constraint is that N must also be divisible by 3.

Here's the key difference from Statement 1: Numbers that are "4 more than a multiple of 8" AND "divisible by 3" repeat every 24 numbers (\(\mathrm{8 × 3 = 24}\)). In our range of 40 numbers, we could have multiple values.

Testing our candidates: Which of \(\mathrm{\{84, 92, 100, 108, 116\}}\) are divisible by 12?

• \(\mathrm{84 ÷ 12 = 7}\) ✓
• \(\mathrm{108 ÷ 12 = 9}\) ✓

We get TWO possible values: 84 or 108.

[STOP - Statement 2 is NOT Sufficient!]

This eliminates choices B and D.

The Answer: A

Statement 1 alone uniquely determines that there are 108 boxes, while Statement 2 alone gives us two possibilities (84 or 108).

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."