Puzzle for July 18, 2023 ( )
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 B with C + F (from eq.3): C + F + C = A + F which becomes 2×C + F = A + F Subtract F from each side of the equation above: 2×C + F - F = A + F - F which makes 2×C = A
Hint #2
In eq.2, replace A with 2×C: E = 2×C + C which makes E = 3×C
Hint #3
In eq.5, substitute A + C for E (from eq.2): A + D = B + A + C - A which becomes eq.5a) A + D = B + C
Hint #4
Substitute A + D for B + C (from eq.5a) into eq.4: A + D = A + F Subtract A from each side of the equation above: A + D - A = A + F - A which makes D = F
Hint #5
Substitute 2×C for A, 3×C for E, C + F for B (from eq.3), and F for D in eq.6: 2×C + C + 3×C = C + F + F + F which becomes 6×C = C + 3×F Subtract C from each side of the above equation: 6×C - C = C + 3×F - C which makes 5×C = 3×F Divide both sides by 3: 5×C ÷ 3 = 3×F ÷ 3 which makes 1⅔×C = F and also makes 1⅔×C = F = D
Hint #6
Substitute 1⅔×C for F in eq.3: B = C + 1⅔×C which makes B = 2⅔×C
Solution
Substitute 2×C for A, 2⅔×C for B, 1⅔×C for D and F, and 3×C for E in eq.1: 2×C + 2⅔×C + C + 1⅔×C + 3×C + 1⅔×C = 36 which simplifies to 12×C = 36 Divide both sides of the above equation by 12: 12×C ÷ 12 = 36 ÷ 12 which means C = 3 making A = 2×C = 2 × 3 = 6 B = 2⅔×C = 2⅔ × 3 = 8 D = F = 1⅔×C = 1⅔ × 3 = 5 E = 3×C = 3 × 3 = 9 and ABCDEF = 683595