Puzzle for February 10, 2022 ( )
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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.
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Hint #1
In eq.5, replace E with A + D (from eq.2): D + A + D – A = A + C which becomes eq.5a) 2×D = A + C
Hint #2
Add C to both sides of eq.3: C + D + C = A – C + C which becomes 2×C + D = A In eq.5a, replace A with 2×C + D: 2×D = 2×C + D + C which becomes 2×D = 3×C + D Subtract D from each side of the equation above: 2×D – D = 3×C + D – D which makes D = 3×C
Hint #3
In eq.5a, substitute (3×C) for D: 2×(3×C) = A + C which becomes 6×C = A + C Subtract C from each side of the above equation: 6×C – C = A + C – C which makes 5×C = A
Hint #4
Substitute 5×C for A, and 3×C for D in eq.2: E = 5×C + 3×C which makes E = 8×C
Hint #5
Substitute 8×C for E in eq.4: B + 8×C = C + F Subtract C from each side of the above equation: B + 8×C – C = C + F – C which becomes eq.4a) B + 7×C = F
Hint #6
Substitute 5×C for A, 8×C for E, and (B + 7×C) for F (from eq.4a) in eq.6: 5×C + 8×C – (B + 7×C) = B – C + (B + 7×C) which becomes 13×C – B – 7×C = 2×B + 6×C which becomes 6×C – B = 2×B + 6×C In the above equation, add B to both sides, and subtract 6×C from both sides: 6×C – B + B – 6×C = 2×B + 6×C + B – 6×C which makes 0 = 3×B which means 0 = B
Hint #7
Substitute 0 for B in eq.4a: 0 + 7×C = F which means 7×C = F
Solution
Substitute 5×C for A, 0 for B, 3×C for D, 8×C for E, and 7×C for F in eq.1: 5×C + 0 + C + 3×C + 8×C + 7×C = 24 which simplifies to 24×C = 24 Divide both sides of the above equation by 24: 24×C ÷ 24 = 24 ÷ 24 which means C = 1 making A = 5×C = 5 × 1 = 5 D = 3×C = 3 × 1 = 3 E = 8×C = 8 × 1 = 8 F = 7×C = 7 × 1 = 7 and ABCDEF = 501387