Puzzle for August 24, 2021 ( )
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, substitute A + C for D + F (from eq.2): B + C = A + A + C which becomes B + C = 2×A + C Subtract C from each side of the above equation: B + C – C = 2×A + C – C which makes eq.5a) B = 2×A
Hint #2
In eq.3, replace B with 2×A (from eq.5a): A – E = 2×A – A which becomes A – E = A Subtract A from each side of the equation above: A – E – A = A – A which makes –E = 0 which means E = 0
Hint #3
In eq.6, replace B with 2×A (from eq.5a): C ÷ D = 2×A ÷ A which becomes C ÷ D = 2 Multiply both sides of the above equation by D: (C ÷ D) × D = 2 × D which makes C = 2×D
Hint #4
In eq.4, substitute 2×D for C, 2×A for B (from eq.5a), and 0 for E: 2×D + D – 2×A = 2×A + 0 which becomes 3×D – 2×A = 2×A Add 2×A to both sides of the above equation: 3×D – 2×A + 2×A = 2×A + 2×A which makes 3×D = 4×A Divide both sides by 4: 3×D ÷ 4 = 4×A ÷ 4 which makes ¾×D = A
Hint #5
Substitute (¾×D) for A in eq.5a: B = 2×(¾×D) which makes B = 1½×D
Hint #6
Substitute ¾×D for A, and 2×D for C in eq.2: D + F = ¾×D + 2×D which becomes D + F = 2¾×D Subtract D from both sides of the equation above: D + F – D = 2¾×D – D which makes F = 1¾×D
Solution
Substitute ¾×D for A, 1½×D for B, 2×D for C, 0 for E, and 1¾×D for F in eq.1: ¾×D + 1½×D + 2×D + D + 0 + 1¾×D = 28 which simplifies to 7×D = 28 Divide both sides of the above equation by 7: 7×D ÷ 7 = 28 ÷ 7 which means D = 4 making A = ¾×D = ¾ × 4 = 3 B = 1½×D = 1½ × 4 = 6 C = 2×D = 2 × 4 = 8 F = 1¾×D = 1¾ × 4 = 7 and ABCDEF = 368407