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9. If we add J to each side, we have ABC+DEF+GHI+J= JJJ+J. Dividing by 9, the left side leaves a zero remainder (since the sum of all digits from 0 to 9 is 45, a multiple of 9) so the right side must also leave a zero remainder; thus, 4xJ must be a multiple of 9, so J=9. Note: This doesn't actually prove that J=9; it could well be that the problem had no solution. However, it's easy to find such solutions, and that completes the proof. |
