Let A3BC and DE2F be four-digit numbers where each letter represents a different digit greater than 3. If the sum of the numbers is 15902, then what is the difference between the values of A and D?

Here's a brief, clear explanation:

1. Set up the addition: A3BC

   • DE2F

   ---

   15902

2. Analyze the units column (C + F): C + F must result in a number ending in 2. Since C and F are digits greater than 3 (minimum 4+5=9, maximum 9+8=17), the only possibility is C + F = 12. This gives a carry-over of 1 to the tens column.

3. Analyze the tens column (B + 2 + carry-over): B + 2 + 1 = B + 3 must result in a number ending in 0. Since B is a digit greater than 3 (minimum 4+3=7, maximum 9+3=12), the only possibility is B + 3 = 10. Therefore, B = 7. This gives a carry-over of 1 to the hundreds column.

4. Analyze the hundreds column (3 + E + carry-over): 3 + E + 1 = 9. Therefore, E = 5. There is no carry-over to the thousands column.

5. Analyze the thousands column (A + D): A + D = 15.

6. Apply constraints:

   • All letters (A, B, C, D, E, F) represent different digits greater than 3.
   • Digits greater than 3 are {4, 5, 6, 7, 8, 9}.
   • We've found B = 7 and E = 5.
   • So, the remaining available digits for A, D, C, F are {4, 6, 8, 9}.

7. Determine A and D: From A + D = 15, and A, D must be distinct digits from {4, 6, 8, 9}:

   • If A = 6, then D = 9. This is a valid pair (both 6 and 9 are in the remaining set).
   • If A = 9, then D = 6. This is also a valid pair.
   • No other combination from {4, 6, 8, 9} sums to 15 (e.g., 4+x cannot be 15, 8+x would require x=7, but 7 is already used by B).

8. Calculate the difference: In both valid cases, the values of A and D are 6 and 9. The difference between them is |9 - 6| = 3.

The final answer is 3.

The final answer is C