AB and CD are 2-digit numbers. Multiplying AB with CD results in a 3-digit number DEF. Adding DEF to another 3-digit number GHI results in 975. Further A, B, C, D, E, F, G, H, I are distinct digits. If E = 0, F = 8, then what is A + B + C equal to?

The problem provides two equations and a set of constraints: 1. AB * CD = DEF 2. DEF + GHI = 975 3. A, B, C, D, E, F, G, H, I are distinct digits. 4. E = 0, F = 8. Step 1: Use the known values E=0, F=8 to determine DEF. DEF = D08. Step 2: Substitute DEF into the second equation. D08 + GHI = 975 Step 3: Determine possible values for D. D must be a digit (0-9) and distinct from E=0 and F=8. Since DEF is a 3-digit number, D cannot be 0. So D can be 1, 2, 3, 4, 5, 6, 7, 9. Let's test these values for D, keeping the distinct digit constraint in mind: * If D = 1, DEF = 108. GHI = 975 - 108 = 867. Digits used: D=1, E=0, F=8, G=8, H=6, I=7. Here, F=8 and G=8. This violates the distinct digit constraint (F and G must be distinct). So D cannot be 1. * If D = 2, DEF = 208. GHI = 975 - 208 = 767. Digits used: D=2, E=0, F=8, G=7, H=6, I=7. Here, G=7 and I=7. This violates the distinct digit constraint. So D cannot be 2. * If D = 3, DEF = 308. GHI = 975 - 308 = 667. Digits used: D=3, E=0, F=8, G=6, H=6, I=7. Here, G=6 and H=6. This violates the distinct digit constraint. So D cannot be 3. * If D = 4, DEF = 408. GHI = 975 - 408 = 567. Digits used: D=4, E=0, F=8, G=5, H=6, I=7. These digits {0, 4, 5, 6, 7, 8} are all distinct. This is a valid set. So, D=4, E=0, F=8, G=5, H=6, I=7. Digits remaining for A, B, C are from {1, 2, 3, 9}. Step 4: Use the first equation AB * CD = DEF with D=4 and DEF=408. AB * C4 = 408. We need to find two 2-digit numbers, one ending in 4, whose product is 408. Let's find the factors of 408: 408 = 2^3 * 3 * 17. Factors include: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, etc. We are looking for two 2-digit factors. Possible pairs: (12, 34) (17, 24) Since CD must end in 4 (C4), the only possibility is CD = 34. If CD = 34, then C=3 and D=4. Consequently, AB = 12, so A=1 and B=2. Step 5: Verify all distinct digit constraints. A=1, B=2, C=3, D=4, E=0, F=8, G=5, H=6, I=7. The set of all digits is {1, 2, 3, 4, 0, 8, 5, 6, 7}. All 9 digits are distinct. This solution is consistent with all conditions. Step 6: Calculate A + B + C. A + B + C = 1 + 2 + 3 = 6. (Brief check for other D values, if D=5, DEF=508. GHI=467. Digits 0,8,5,4,6,7 are distinct. AB*C5=508. Factors of 508 are 1,2,4,127,254,508. No two 2-digit factors. So D cannot be 5. Similarly, other D values will lead to contradictions.) The final answer is 6. The final answer is A) 6