In the two-digit integers 3square and 2triangle, the symbols square and triangle represent different digits, and the product \((3\square)(2\triangle)\...

GMAT Data Sufficiency : (DS) Questions

Source: Official Guide

Data Sufficiency

DS-Basics

HARD

...

Notes

Post a Query

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?

The sum of \(\square\) and \(\triangle\) is 10.
The product of \(\square\) and \(\triangle\) is 24

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Solution

Understanding the Question

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:

3□ represents a number from 30-39 (like 32, 35, 38...)
2△ represents a number from 20-29 (like 21, 24, 27...)
Their product equals 864

What we need to determine: Can we find a unique value for □?

Given information:

3□ and 2△ are two-digit integers
□ and △ are different single digits (0-9)
\((3□)(2△) = 864\)

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:

\(864 ÷ 30 = 28.8\) (not a whole number)
\(864 ÷ 31 = 27.87...\) (not a whole number)
\(864 ÷ 32 = 27\) ✓ (perfect! 27 starts with 2)
\(864 ÷ 33 = 26.18...\) (not a whole number)
\(864 ÷ 34 = 25.41...\) (not a whole number)
\(864 ÷ 35 = 24.69...\) (not a whole number)
\(864 ÷ 36 = 24\) ✓ (perfect! 24 starts with 2)
\(864 ÷ 37 = 23.35...\) (not a whole number)
\(864 ÷ 38 = 22.74...\) (not a whole number)
\(864 ÷ 39 = 22.15...\) (not a whole number)

So we have exactly two possibilities:

\((32, 27)\) giving \(□ = 2, △ = 7\)
\((36, 24)\) giving \(□ = 6, △ = 4\)

Both pairs satisfy our "different digits" constraint (\(2 ≠ 7\) and \(6 ≠ 4\)).

Sufficiency means: Finding exactly one value for □.

Analyzing Statement 1

Statement 1 tells us: \(□ + △ = 10\)

Let's check our two possibilities:

If \(□ = 2\) and \(△ = 7: 2 + 7 = 9\) ✗
If \(□ = 6\) and \(△ = 4: 6 + 4 = 10\) ✓

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.

Analyzing Statement 2

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

Statement 2 tells us: \(□ × △ = 24\)

Let's check our two possibilities again:

If \(□ = 2\) and \(△ = 7: 2 × 7 = 14\) ✗
If \(□ = 6\) and \(△ = 4: 6 × 4 = 24\) ✓

[STOP - Statement 2 is Sufficient!]

Statement 2 is sufficient.

This eliminates answer choices A, C, and E.

The Answer: D

Both statements independently lead us to the unique value \(□ = 6\).

Answer Choice D: "Each statement alone is sufficient."

Quick verification: \(36 × 24 = 864\) ✓

Answer Choices Explained

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Rate this Solution

Tell us what you think about this solution

...

Forum Discussions

Start a new discussion

Post

Previous Attempts

Loading attempts...