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In the two-digit integers \(3\square\) and \(2\triangle\), the symbols \(\square\) and \(\triangle\) represent different digits, and the product \((3\square)(2\triangle)\) is equal to 864. What digit does \(\square\) represent?
We need to find the value of □ in the product \((3□)(2△) = 864\), where 3□ and 2△ are two-digit numbers and □ and △ are different digits.
Let me clarify what this means:
What we need to determine: Can we find a unique value for □?
Given information:
Key insight: Since we're dealing with limited possibilities (only 10 choices each for □ and △), we can find which pairs of two-digit numbers multiply to 864. Let's systematically check what divides 864:
So we have exactly two possibilities:
Both pairs satisfy our "different digits" constraint (\(2 ≠ 7\) and \(6 ≠ 4\)).
Sufficiency means: Finding exactly one value for □.
Statement 1 tells us: \(□ + △ = 10\)
Let's check our two possibilities:
Only the pair (36, 24) satisfies Statement 1, so □ must equal 6.
[STOP - Statement 1 is Sufficient!]
Statement 1 is sufficient.
This eliminates answer choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: \(□ × △ = 24\)
Let's check our two possibilities again:
[STOP - Statement 2 is Sufficient!]
Statement 2 is sufficient.
This eliminates answer choices A, C, and E.
Both statements independently lead us to the unique value \(□ = 6\).
Answer Choice D: "Each statement alone is sufficient."
Quick verification: \(36 × 24 = 864\) ✓