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