Puzzle for August 23, 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.5, replace E with D + F (from eq.2): F – D = D + F – F which becomes F – D = D Add D to both sides of the above equation: F – D + D = D + D which makes F = 2×D
Hint #2
In eq.2, replace F with 2×D: E = D + 2×D which makes E = 3×D
Hint #3
In eq.4, replace F with 2×D: D – 2×D = B – D which becomes –D = B – D Add D to both sides of the equation above: –D + D = B – D + D which makes 0 = B
Hint #4
eq.6 may be written as: E = (A + C + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + C + D) ÷ 3 which becomes 3×E = A + C + D which may be written as eq.6a) 3×E = A + D + C
Hint #5
In eq.6a, substitute (3×D) for E, and C for A + D (from eq.3): 3×(3×D) = C + C which becomes 9×D = 2×C Divide both sides of the above equation by 2: 9×D ÷ 2 = 2×C ÷ 2 which makes 4½×D = C
Hint #6
Substitute 4½×D for C in eq.3: 4½×D = A + D Subtract D from each side of the equation above: 4½×D – D = A + D – D which makes 3½×D = A
Solution
Substitute 3½×D for A, 0 for B, 4½×D for C, 3×D for E, and 2×D for F in eq.1: 3½×D + 0 + 4½×D + D + 3×D + 2×D = 28 which simplifies to 14×D = 28 Divide both sides of the above equation by 14: 14×D ÷ 14 = 28 ÷ 14 which means D = 2 making A = 3½×D = 3½ × 2 = 7 C = 4½×D = 4½ × 2 = 9 E = 3×D = 3 × 2 = 6 F = 2×D = 2 × 2 = 4 and ABCDEF = 709264