Puzzle for January 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 E with B + D (from eq.2): A = D + B + D which becomes eq.4a) A = 2×D + B
Hint #2
In eq.3, replace A with 2×D + B (from eq.4a): C = 2×D + B + D which becomes eq.3a) C = 3×D + B
Hint #3
In eq.5, substitute 3×D + B for C (from eq.3a): F = B + 3×D + B which becomes eq.5a) F = 2×B + 3×D
Hint #4
Substitute B + D for E (from eq.2), and 2×B + 3×D for F (from eq.5a) in eq.6: B + B + D = D + 2×B + 3×D - B which becomes 2×B + D = 4×D + B Subtract D and B from each side of the above equation: 2×B + D - D - B = 4×D + B - D - B which makes B = 3×D
Hint #5
Substitute 3×D for B in eq.4a: A = 2×D + 3×D which makes A = 5×D
Hint #6
Substitute 3×D for B in eq.3: C = 3×D + 3×D which makes C = 6×D
Hint #7
Substitute 3×D for B in eq.2: E = 3×D + D which makes E = 4×D
Hint #8
Substitute (3×D) for B in eq.5a: F = 2×(3×D) + 3×D which becomes F = 6×D + 3×D which makes F = 9×D
Solution
Substitute 5×D for A, 3×D for B, 6×D for C, 4×D for E, and 9×D for F in eq.1: 5×D + 3×D + 6×D + D + 4×D + 9×D = 28 which simplifies to 28×D = 28 Divide both sides of the above equation by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 3×D = 3 × 1 = 3 C = 6×D = 6 × 1 = 6 E = 4×D = 4 × 1 = 4 F = 9×D = 9 × 1 = 9 and ABCDEF = 536149