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Each dinner at Restaurant X includes a pair of side dishes, which a customer chooses by selecting any one side dish from list A and any one side dish from list B. There are a total of 18 possible pairs of side dishes, and no side dish appears on both lists. If there are fewer side dishes on list A than on list B, how many side dishes are on list B ?
Let's clarify what we're looking for: We need to find the exact number of side dishes on list B.
Given Information:
Key Mathematical Relationship:
Since total pairs = \(\mathrm{(dishes\,on\,A)} \times \mathrm{(dishes\,on\,B)} = 18\), we need to find the specific value of dishes on list B.
Critical Insight:
The number 18 has limited factor pairs: (1,18), (2,9), (3,6), (6,3), (9,2), (18,1). Since \(\mathrm{A} < \mathrm{B}\), we only consider pairs where the first number is smaller: (1,18), (2,9), and (3,6). This gives us exactly three possibilities for list B: 18, 9, or 6 dishes.
Statement 1: There are a total of 9 side dishes on the two lists.
This means \(\mathrm{A} + \mathrm{B} = 9\).
Let's check which factor pair of 18 satisfies this constraint:
Only one possibility works! Therefore, list B must have exactly 6 dishes.
[STOP - Statement 1 is SUFFICIENT!]
This eliminates choices B, C, and E.
Important: Now we forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: There are a total of 3 side dishes on list A.
This directly tells us \(\mathrm{A} = 3\).
Since \(\mathrm{A} \times \mathrm{B} = 18\) and \(\mathrm{A} = 3\):
\(3 \times \mathrm{B} = 18\)
\(\mathrm{B} = 18 \div 3 = 6\)
We can determine exactly one value: List B has 6 dishes.
[STOP - Statement 2 is SUFFICIENT!]
This eliminates choices A, C, and E.
Both statements independently allow us to determine that list B has exactly 6 dishes.
Answer Choice D: "Each statement alone is sufficient."