Puzzle for September 14, 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
Add C and D to both sides of eq.5: C + D + C + D = A - C - D + F + C + D which becomes eq.5a) 2×C + 2×D = A + F Add C and D to both sides of eq.3: C - D + C + D = A - C + C + D which becomes eq.3a) 2×C = A + D
Hint #2
In eq.5a, replace 2×C with A + D (from eq.3a): A + D + 2×D = A + F which becomes A + 3×D = A + F Subtract A from each side of the equation above: A + 3×D - A = A + F - A which makes 3×D = F
Hint #3
In eq.4, replace F with 3×D: 3×D - D = C - 3×D which becomes 2×D = C - 3×D Add 3×D to both sides of the above equation: 2×D + 3×D = C - 3×D + 3×D which becomes eq.4a) 5×D = C
Hint #4
In eq.3a, substitute (5×D) for C: 2×(5×D) = A + D which becomes 10×D = A + D Subtract D from each side of the above equation: 10×D - D = A + D - D which makes 9×D = A
Hint #5
Substitute 9×D for A, D + E for B (from eq.2), 5×D for C, and 3×D for F in eq.6: 9×D + D + E - E = 5×D + E - 3×D which becomes 10×D = 2×D + E Subtract 2×D from both sides of the equation above: 10×D - 2×D = 2×D + E - 2×D which makes 8×D = E
Hint #6
Substitute 8×D for E in eq.2: B = D + 8×D which makes B = 9×D
Solution
Substitute 9×D for A and B, 5×D for C, 8×D for E, and 3×D for F in eq.1: 9×D + 9×D + 5×D + D + 8×D + 3×D = 35 which simplifies to 35×D = 35 Divide both sides of the above equation by 35: 35×D ÷ 35 = 35 ÷ 35 which means D = 1 making A = B = 9×D = 9 × 1 = 9 C = 5×D = 5 × 1 = 5 E = 8×D = 8 × 1 = 8 F = 3×D = 3 × 1 = 3 and ABCDEF = 995183