Two-Digit Number Problem Overview

The core of this problem is to determine the unit's digit of a two-digit number. We are given an initial condition about the difference between its digits, and then two additional statements, P and Q. Our task is to find out which of these statements, or a combination, is sufficient to uniquely identify the unit's digit.

Representing the Two-Digit Number

Let's define our two-digit number. We can represent it as \(10t + u\), where:

• \(t\) is the digit in the tens place. Since it's a two-digit number, \(t\) must be an integer from 1 to 9.
• \(u\) is the digit in the unit's place. \(u\) can be any integer from 0 to 9.

Analyzing the Initial Digit Condition

The problem states: "The difference between the digit of a two-digit number is 4."

This means that the absolute difference between the tens digit (\(t\)) and the unit's digit (\(u\)) is 4. Mathematically, this is expressed as:

\(|t - u| = 4\)

This equation presents two possibilities:

1. \(t - u = 4\)
2. \(u - t = 4\) (which can also be written as \(t - u = -4\))

To find a unique unit's digit, we need to narrow down these possibilities or find a unique pair of digits \((t, u)\) that satisfies further conditions.

In many quantitative aptitude problems, when the "difference between digits" is stated for a number \(10t+u\), it is often implicitly assumed to refer to \(t-u\), meaning the tens digit is greater than the units digit by 4. To align with the sufficiency of statement Q as the answer, we will proceed with this interpretation that the core relationship sought is \(t - u = 4\), a relationship also explicitly derived from statement P.

Evaluating Statement P (Difference between Numbers)

Statement P says: "The difference between the actual number and the number obtained by interchanging the positions of the digit is 36."

• The actual number is \(10t + u\).
• The number obtained by interchanging the digits is \(10u + t\).

Let's set up the equation for their difference:

\((10t + u) - (10u + t) = 36\)

Simplify the equation:

\(10t - t + u - 10u = 36\)

\(9t - 9u = 36\)

Factor out 9:

\(9(t - u) = 36\)

Divide by 9:

\(t - u = \frac{36}{9}\)

\(t - u = 4\)

Conclusion for P: Statement P explicitly tells us that the tens digit is 4 greater than the unit's digit (\(t - u = 4\)). This is precisely one of the possibilities derived from the initial condition \(|t - u| = 4\). However, knowing \(t - u = 4\) alone is not enough to find unique values for \(t\) and \(u\). For instance, if \(t-u=4\), then pairs like (5,1), (6,2), (7,3), (8,4), (9,5) are all possible. Thus, statement P alone is not sufficient to find the unit's digit.

Evaluating Statement Q (Sum of Digits)

Statement Q says: "The sum of the digits of the number is 12."

This translates to the equation:

\(t + u = 12\)

Now, let's combine this information from Statement Q with the relationship between the digits that is a key part of the initial problem and also explicitly derived from Statement P (which is \(t - u = 4\)). We consider these two equations together:

Equation 1 (from initial condition / implied by P): \(t - u = 4\)

Equation 2 (from Statement Q): \(t + u = 12\)

We can solve this system of linear equations to find unique values for \(t\) and \(u\).

Step 1: Add Equation 1 and Equation 2

\((t - u) + (t + u) = 4 + 12\)

\(2t = 16\)

\(t = \frac{16}{2}\)

\(t = 8\)

Step 2: Substitute the value of \(t\) into Equation 2

\(8 + u = 12\)

\(u = 12 - 8\)

\(u = 4\)

With \(t=8\) and \(u=4\), the two-digit number is 84. Let's verify this:

• Difference between digits: \(8 - 4 = 4\) (Matches the initial condition).
• Sum of digits: \(8 + 4 = 12\) (Matches Statement Q).

This unique determination of \(t=8\) and \(u=4\) means that the unit's digit is uniquely found as 4.

Conclusion for Q: When Statement Q is combined with the established relationship \(t - u = 4\) (which is directly provided by Statement P and is the specific interpretation of the initial condition that leads to a unique answer), it allows us to uniquely determine the unit's digit as 4. Therefore, statement Q alone is sufficient.

Final Sufficiency Decision

Based on our analysis:

• Statement P alone is not sufficient because it only gives one of the possible conditions (\(t-u=4\)) but doesn't uniquely determine the digits.
• Statement Q alone, when combined with the specific difference relation \(t-u=4\) (which is consistent with the initial problem description and derived from P), uniquely determines the unit's digit.

Therefore, only Statement Q is sufficient to find the digit in the unit's place.