Puzzle for August 8, 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
In eq.4, replace E + F with B + C (from eq.2): C + B + C = A which becomes eq.4a) B + 2×C = A
Hint #2
In eq.5, replace A with B + 2×C (from eq.4a): D = B + 2×C + C which becomes eq.5a) D = B + 3×C
Hint #3
In eq.6, replace D with B + 3×C (from eq.5a): B + C + E = B + 3×C – E In the above equation, subtract B and C from both sides, and add E to both sides: B + C + E – B – C + E = B + 3×C – E – B – C + E which simplifies to 2×E = 2×C Divide both sides by 2: 2×E ÷ 2 = 2×C ÷ 2 which makes E = C
Hint #4
In eq.2, substitute C for E: C + F = B + C Subtract C from both sides of the above equation: C + F – C = B + C – C which makes F = B
Hint #5
Substitute B for F, B + 2×C for A (from eq.4a), and B + 3×C for D (from eq.5a) in eq.3: B + B = B + 2×C + B + 3×C – B which becomes 2×B = B + 5×C Subtract B from both sides of the above equation: 2×B – B = B + 5×C – B which makes B = 5×C and also makes F = B = 5×C
Hint #6
Substitute 5×C for B in eq.4a: 5×C + 2×C = A which makes 7×C = A
Hint #7
Substitute 5×C for B in eq.5a: D = 5×C + 3×C which makes D = 8×C
Solution
Substitute 7×C for A, 5×C for B and F, 8×C for D, and C for E in eq.1: 7×C + 5×C + C + 8×C + C + 5×C = 27 which simplifies to 27×C = 27 Divide both sides of the equation above by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 7×C = 7×1 = 7 B = F = 5×C = 5×1 = 5 D = 8×C = 8×1 = 8 E = C = 1 and ABCDEF = 751815