Puzzle for September 12, 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.2, replace D with A + C (from eq.3): A + E = C + A + C which becomes A + E = 2×C + A Subtract A from each side of the equation above: A + E - A = 2×C + A - A which makes E = 2×C
Hint #2
In eq.4, replace E with 2×C: C + 2×C = B - C which becomes 3×C = B - C Add C to both sides of the above equation: 3×C + C = B - C + C which makes 4×C = B
Hint #3
In eq.5, substitute 4×C for B: 4×C - A = A - 4×C + C which becomes 4×C - A = A - 3×C Add A and 3×C to both sides of the equation above: 4×C - A + A + 3×C = A - 3×C + A + 3×C which makes 7×C = 2×A Divide both sides by 2: 7×C ÷ 2 = 2×A ÷ 2 which makes 3½×C = A
Hint #4
Substitute 3½×C for A in eq.3: D = 3½×C + C which makes D = 4½×C
Hint #5
Substitute 4½×C for D, and 2×C for E in eq.6: F = 4½×C - 2×C which makes F = 2½×C
Solution
Substitute 3½×C for A, 4×C for B, 4½×C for D, 2×C for E, and 2½×C for F in eq.1: 3½×C + 4×C + C + 4½×C + 2×C + 2½×C = 35 which simplifies to 17½×C = 35 Divide both sides of the above equation by 17½: 17½×C ÷ 17½ = 35 ÷ 17½ which means C = 2 making A = 3½×C = 3½ × 2 = 7 B = 4×C = 4 × 2 = 8 D = 4½×C = 4½ × 2 = 9 E = 2×C = 2 × 2 = 4 F = 2½×C = 2½ × 2 = 5 and ABCDEF = 782945