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 ?
- (a) 6 ✓ UPSC's answer
- (b) 7
- (c) 8
- (d) 9
Why the answer is (a)
• Since E = 0 and F = 8, the product DEF is D08, which is a 3-digit number ending in 8.
• The sum DEF + GHI = 975 implies D08 + GHI = 975, so GHI = 975 - D08.
• For GHI to be a 3-digit number with distinct digits, D must be 1, 2, 3, 4, 5, 6, 7, or 8 (D cannot be 0 or 9 as DEF is 3-digit and GHI must be positive 3-digit).
• Testing values: If D=1, DEF=108, GHI=867. Digits {1,0,8,8,6,7} have duplicate 8, invalid.
• If D=2, DEF=208, GHI=767. Digits {2,0,8,7,6,7} have duplicate 7, invalid.
• If D=3, DEF=308, GHI=667. Digits {3,0,8,6,6,7} have duplicate 6, invalid.
• If D=4, DEF=408, GHI=567. Digits {4,0,8,5,6,7} are all distinct. Thus, DEF=408 and GHI=567.
• Now, AB * CD = 408. Since CD is a 2-digit number and AB is a 2-digit number, and their product is 408, we check factors of 408. 408 = 12 * 34. Here AB=12, CD=34 (or vice versa). Digits A=1, B=2, C=3, D=4. All digits A,B,C,D,E,F,G,H,I are {1,2,3,4,0,8,5,6,7}, which are distinct.
• Therefore, A + B + C = 1 + 2 + 3 = 6.
Why the other options are wrong
- (b) 7
- The sum A + B + C is 6, not 7, based on the unique factorization of 408 into two 2-digit numbers with distinct digits.
- (c) 8
- The sum A + B + C is 6, not 8, as derived from the valid combination AB=12 and CD=34.
- (d) 9
- The sum A + B + C is 6, not 9, because the only valid distinct-digit solution for the product 408 yields A=1, B=2, C=3.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.