Puzzle for May 31, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.3, replace B with A + C (from eq.1): A – C = A + C – A which becomes A – C = C Add C to both sides of the above equation: A – C + C = C + C which makes A = 2×C
Hint #2
In eq.1, replace A with 2×C: B = 2×C + C which makes B = 3×C
Hint #3
In eq.6, substitute 3×C for B, and 2×C for A: F ÷ 3×C = 3×C ÷ 2×C which becomes F ÷ 3×C = 1½ Multiply both sides of the above equation by 3×C: (F ÷ 3×C) × 3×C = 1½ × 3×C which makes F = 4½×C
Hint #4
Substitute 4½×C for F, 2×C for A, and 3×C for B in eq.4: D + 4½×C = 2×C + 3×C + C which becomes D + 4½×C = 6×C Subtract 4½×C from each side of the equation above: D + 4½×C – 4½×C = 6×C – 4½×C which makes D = 1½×C
Hint #5
Substitute 3×C for B, and 1½×C for D in eq.2: E = 3×C + 1½×C which makes E = 4½×C
Solution
Substitute 3×C for B, 1½×C for D, and 4½×C for E and F in eq.5: 3×C × 1½×C = 4½×C + 4½×C which becomes 4½×C² = 9×C Since C ≠ 0 (from eq.6a), divide both sides of the above equation by 4½×C: 4½×C² ÷ 4½×C = 9×C ÷ 4½×C which makes C = 2 and makes A = 2×C = 2 × 2 = 4 B = 3×C = 3 × 2 = 6 D = 1½×C = 1½ × 2 = 3 E = F = 4½×C = 4½ × 2 = 9 and ABCDEF = 462399