Puzzle for February 18, 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.2, replace F with A + C (from eq.4): A + A + C = C + E which becomes 2×A + C = C + E Subtract C from both sides of the above equation: 2×A + C - C = C + E - C which makes 2×A = E
Hint #2
Replace E with 2×A, and F with A + C (from eq.4) in eq.5: C + D = A + 2×A + A + C which becomes C + D = 4×A + C Subtract C from both sides of the above equation: C + D - C = 4×A + C - C which makes D = 4×A
Hint #3
Substitute 4×A for D, 2×A for E, and A + C for F (from eq.4) into eq.6: 4×A + 2×A = C + A + C which becomes 6×A = A + 2×C Subtract A from each side of the equation above: 6×A - A = A + 2×C - A which becomes 5×A = 2×C Divide both sides by 2: 5×A ÷ 2 = 2×C ÷ 2 which means 2½×A = C
Hint #4
In eq.4, substitute 2½×A for C: F = A + 2½×A which makes F = 3½×A
Hint #5
Substitute 4×A for D, 2½×A for C, 2×A for E, and 3½×A for F in eq.3: B + 4×A = 2½×A + 2×A + 3½×A which becomes B + 4×A = 8×A Subtract 4×A from both sides: B + 4×A - 4×A = 8×A - 4×A which makes B = 4×A
Solution
Substitute 4×A for B and D, 2½×A for C, 2×A for E, and 3½×A for F in eq.1: A + 4×A + 2½×A + 4×A + 2×A + 3½×A = 34 which simplifies to 17×A = 34 Divide each side by 17: 17×A ÷ 17 = 34 ÷ 17 which means A = 2 making B = D = 4×A = 4 × 2 = 8 C = 2½×A = 2½ × 2 = 5 E = 2×A = 2 × 2 = 4 F = 3½×A = 3½ × 2 = 7 and ABCDEF = 285847