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 B?

Correct Option (2)

The given numbers are A3BC and DE2F, where each letter (A, B, C, D, E, F) represents a different digit greater than 3. The sum of these four-digit numbers is 15902.

To determine the values of the letters, we analyze the sum column by column, starting from the units place:

• Units Place: C + F = 2. Since C and F are digits greater than 3, their minimum sum is 4 + 5 = 9. Therefore, C + F must result in a carry-over. The only possibility for the units digit of the sum to be 2 is if C + F = 12. This generates a carry-over of 1 to the tens place.
• Tens Place: B + 2 + (carry-over 1) = 0. This simplifies to B + 3 = 0. Since B is a digit greater than 3, its minimum value is 4. Therefore, B + 3 must result in a carry-over. The only possibility for the units digit of the sum to be 0 is if B + 3 = 10. This generates a carry-over of 1 to the hundreds place. From B + 3 = 10, we deduce B = 7.
• Hundreds Place: 3 + E + (carry-over 1) = 9. This simplifies to 4 + E = 9. From this, we deduce E = 5. There is no carry-over to the thousands place as 4 + 5 = 9.
• Thousands Place: A + D = 15. Since there was no carry-over from the hundreds place, and the sum is 15902, the sum of the thousands digits (A + D) must be 15.

At this stage, we have determined B = 7 and E = 5.

The digits A, B, C, D, E, F must be distinct and greater than 3. The available digits greater than 3 are {4, 5, 6, 7, 8, 9}.

Digits already assigned: B=7, E=5.

Remaining available digits for C, F, A, D: {4, 6, 8, 9}.

• Considering C + F = 12: From the remaining available digits {4, 6, 8, 9}, the only pairs that sum to 12 are (4, 8) or (8, 4). Thus, {C, F} = {4, 8}.
• Digits used for C and F: 4, 8.
• Remaining available digits for A and D: {6, 9}.
• Considering A + D = 15: From the remaining available digits {6, 9}, the only pair that sums to 15 is (6, 9) or (9, 6). Thus, {A, D} = {6, 9}.

The question asks for the difference between the values of A and B (A - B).

We have B = 7.

A can be 6 or 9.

• If A = 6, then A - B = 6 - 7 = -1. This is not a positive option provided.
• If A = 9, then A - B = 9 - 7 = 2. This matches option 2.

All assigned digits (A=9, B=7, C=4, D=6, E=5, F=8) are distinct and greater than 3, satisfying all conditions of the problem. For instance, 9374 + 6528 = 15902.

Therefore, the difference between A and B is 2.

Incorrect Options:

The analysis establishes that B must be 7. For the difference A - B to be 1 (Option 1), A would need to be 8. However, if A=8, then D would have to be 7 (since A+D=15). This is not permissible because B is already 7, and all letters must represent different digits. Hence, A - B cannot be 1.

For the difference A - B to be 3 (Option 3), A would need to be 10 (since B=7). A digit cannot be 10. Therefore, A - B cannot be 3.

For the difference A - B to be 4 (Option 4), A would need to be 11 (since B=7). A digit cannot be 11. Therefore, A - B cannot be 4.

The only difference for A - B that is consistent with all problem constraints is 2.