Puzzle for January 30, 2019 ( )
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
Add C to both sides of eq.6: D - C + C = A + C + C which becomes eq.6a) D = A + 2×C
Hint #2
In eq.4, replace D with A + 2×C (from eq.6a): A + A + 2×C = F which becomes eq.4a) 2×A + 2×C = F
Hint #3
In eq.3, substitute A + 2×C for D (from eq.6a), and 2×A + 2×C for F (from eq.4a): A + E = C + A + 2×C + 2×A + 2×C which becomes A + E = 3×A + 5×C Subtract A from both sides of the above equation: A + E - A = 3×A + 5×C - A which simplifies to eq.3a) E = 2×A + 5×C
Hint #4
In eq.2, substitute A + 2×C for D (from eq.6a), and 2×A + 2×C for F (from eq.4a): B + C = A + 2×C + 2×A + 2×C which becomes B + C = 3×A + 4×C Subtract C from both sides of the above equation: B + C - C = 3×A + 4×C - C which becomes eq.2a) B = 3×A + 3×C
Hint #5
Substitute 3×A + 3×C for B (from eq.2a), and 2×A + 5×C for E (from eq.3a) in eq.5: A + 3×A + 3×C - C = C + 2×A + 5×C which becomes 4×A + 2×C = 2×A + 6×C Subtract both 2×A and 2×C from each side of the above equation: 4×A + 2×C - 2×A - 2×C = 2×A + 6×C - 2×A - 2×C which simplifies to 2×A = 4×C Divide each side by 2: 2×A ÷ 2 = 4×C ÷ 2 which makes A = 2×C
Hint #6
Substitute (2×C) for A in eq.2a: B = 3×(2×C) + 3×C which becomes B = 6×C + 3×C which makes B = 9×C
Hint #7
Substitute 2×C for A in eq.6a: D = 2×C + 2×C which makes D = 4×C
Hint #8
Substitute (2×C) for A in eq.3a: E = 2×(2×C) + 5×C which becomes E = 4×C + 5×C which means E = 9×C
Hint #9
Substitute (2×C) for A in eq.4a: 2×(2×C) + 2×C = F which becomes 4×C + 2×C = F which makes 6×C = F
Solution
Substitute 2×C for A, 9×C B and E, 4×C for D, and 6×C for F in eq.1: 2×C + 9×C + C + 4×C + 9×C + 6×C = 31 which becomes 31×C = 31 Divide each side by 31: 31×C ÷ 31 = 31 ÷ 31 which means C = 1 making A = 2×C = 2 × 1 = 2 B = E = 9×C = 9 × 1 = 9 D = 4×C = 4 × 1 = 4 F = 6×C = 6 × 1 = 6 and ABCDEF = 291496