When all the boxes in a warehouse were arranged in stacks of 8, there were 4 boxes left over. If...
GMAT Data Sufficiency : (DS) Questions
When all the boxes in a warehouse were arranged in stacks of 8, there were 4 boxes left over. If there were more than 80 but fewer than 120 boxes in the warehouse, how many boxes were there?
- If all the boxes in the warehouse had been arranged ins tacks of 9, there would have been no boxes left over.
- If all the boxes in the warehouse had been arranged in stacks of 12, there would have been no boxes left over
Understanding the Question
We need to find the exact number of boxes in the warehouse.
Given Information
- When boxes are arranged in stacks of 8, there are 4 left over
- This means: \(\mathrm{Number\ of\ boxes} = 8k + 4\) (where k is a positive integer)
- Total boxes is between 80 and 120: \(80 < \mathrm{boxes} < 120\)
What We Need to Determine
We need to find ONE specific value for the number of boxes. Let's see what values are possible:
- From \(80 < 8k + 4 < 120\), we get \(76 < 8k < 116\)
- Dividing by 8: \(9.5 < k < 14.5\)
- Since k must be a whole number: k can be 10, 11, 12, 13, or 14
- Therefore, possible values are: 84, 92, 100, 108, or 116 boxes
For sufficiency, we need information that narrows these five possibilities down to exactly ONE value.
Analyzing Statement 1
Statement 1: If boxes were arranged in stacks of 9, there would be no remainder.
This means the number must be divisible by 9. Let's check which of our possible values {84, 92, 100, 108, 116} are divisible by 9:
Quick divisibility test for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- 84: Sum = 8 + 4 = 12 → NOT divisible by 9
- 92: Sum = 9 + 2 = 11 → NOT divisible by 9
- 100: Sum = 1 + 0 + 0 = 1 → NOT divisible by 9
- 108: Sum = 1 + 0 + 8 = 9 → Divisible by 9 ✓
- 116: Sum = 1 + 1 + 6 = 8 → NOT divisible by 9
Only 108 is divisible by 9.
Since Statement 1 narrows our answer to exactly one value (108 boxes), it is sufficient.
[STOP - Sufficient!]
This eliminates choices B, C, and E. The answer must be A or D.
Analyzing Statement 2
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: If boxes were arranged in stacks of 12, there would be no remainder.
This means the number must be divisible by 12. Let's check our possible values {84, 92, 100, 108, 116}:
- \(84 ÷ 12 = 7\) → Divisible by 12 ✓
- \(92 ÷ 12 = 7.67...\) → NOT divisible by 12
- \(100 ÷ 12 = 8.33...\) → NOT divisible by 12
- \(108 ÷ 12 = 9\) → Divisible by 12 ✓
- \(116 ÷ 12 = 9.67...\) → NOT divisible by 12
We have TWO values that work: 84 and 108.
Since we can't determine which one is the actual number of boxes, Statement 2 alone is NOT sufficient.
This eliminates choices B and D.
The Answer: A
- Statement 1 alone → exactly one value (108) → Sufficient
- Statement 2 alone → two possible values (84 or 108) → Not sufficient
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."