Puzzle for October 31, 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 D to both sides of eq.5: B + D + D = E - D + D which becomes eq.5a) B + 2×D = E
Hint #2
In eq.6, replace C with D + F (from eq.2), and E with B + 2×D (from eq.5a): D + F + B + 2×D - B = B + F which becomes 3×D + F = B + F Subtract B from both sides of the equation above: 3×D + F - F = B + F - F which makes 3×D = B
Hint #3
In eq.5a, replace B with 3×D: 3×D + 2×D = E which makes 5×D = E
Hint #4
In eq.3, substitute 5×D for E: 5×D = A + D Subtract D from each side of the equation above: 5×D - D = A + D - D which makes 4×D = A
Hint #5
Substitute 4×D for A, and 3×D for B in eq.4: F = 4×D + 3×D + D which makes F = 8×D
Hint #6
Substitute 8×D for F in eq.2: C = D + 8×D which makes C = 9×D
Solution
Substitute 4×D for A, 3×D for B, 9×D for C, 5×D for E, and 8×D for F in eq.1: 4×D + 3×D + 9×D + D + 5×D + 8×D = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 3×D = 3 × 1 = 3 C = 9×D = 9 × 1 = 9 E = 5×D = 5 × 1 = 5 F = 8×D = 8 × 1 = 8 and ABCDEF = 439158