The difference between the digits of a two-digit number is 4. What is the digit in the unit's place?

To find out the answer, which of the information given in the statements P and Q is/are sufficient:

P: The difference between the number and the number obtained by interchanging the positions of the digits is 36.

Q: The sum of the digits of that number is 12.

Both P and Q are needed

The correct answer is option 3: Both P and Q are needed

The phrase "difference between digits is 4" means $|x - y| = 4$.

• Possibility A: $x - y = 4$ (e.g., 84, 73)

• Possibility B: $y - x = 4$ (e.g., 48, 37)

  At this stage, we do not know if the unit's digit is the smaller or larger one.

Statement P : "The difference between the number ($10x + y$) and its reverse ($10y + x$) is 36."

This confirms that the tens digit is larger than the unit's digit. However, it does not provide the specific values (the number could be 51, 62, 73, 84, or 95).

So P alone is insufficient.

Statement Q : "The sum of the digits is 12." : $\mathbf{x + y = 12}$

If we use only Q with the original prompt ($|x - y| = 4$):

• If $x - y = 4$, the number is 84 (Unit's digit = 4).

• If $y - x = 4$, the number is 48 (Unit's digit = 8).

Because we have two different possible unit digits (4 and 8), Statement Q alone cannot give a unique answer. Q alone is insufficient.

Combining P and Q: When we use both: 

• From P: We know $x > y$ (the tens digit is larger).

• From Q: We know the digits must be 8 and 4 (since $8+4=12$ and $8-4=4$).

  With the order fixed by P, the number must be 84. The unit's digit is uniquely identified as 4.