Puzzle for September 19, 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.6: E - D + D = C + D + D which becomes eq.6a) E = C + 2×D
Hint #2
In eq.5, replace E with C + 2×D (from eq.6a): C + 2×D - C = C - D which becomes 2×D = C - D Add D to both sides of the above equation: 2×D + D = C - D + D which makes 3×D = C
Hint #3
In eq.6a, replace C with 3×D: E = 3×D + 2×D which makes E = 5×D
Hint #4
In eq.3, replace C with 3×D: A = 3×D + D which makes A = 4×D
Hint #5
In eq.4, substitute 4×D for A, and 5×D for E: F = 4×D + 5×D which makes F = 9×D
Hint #6
Substitute 9×D for F in eq.2: 9×D = B + D Subtract D from each side of the equation above: 9×D - D = B + D - D which makes 8×D = B
Solution
Substitute 4×D for A, 8×D for B, 3×D for C, 5×D for E, and 9×D for F in eq.1: 4×D + 8×D + 3×D + D + 5×D + 9×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 = 8×D = 8 × 1 = 8 C = 3×D = 3 × 1 = 3 E = 5×D = 5 × 1 = 5 F = 9×D = 9 × 1 = 9 and ABCDEF = 483159