Puzzle for January 2, 2023 ( )
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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.
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Hint #1
In eq.4, replace C with A + D (from eq.3): D - A = A + D - D which becomes D - A = A Add A to both sides of the above equation: D - A + A = A + A which makes D = 2×A
Hint #2
Add D to both sides of eq.4: D - A + D = C - D + D which becomes 2×D - A = C In the above equation, replace D with (2×A): 2×(2×A) - A = C which becomes 4×A - A = C which becomes 3×A = C
Hint #3
In eq.2, replace D with 2×A: B - A = A + 2×A which becomes B - A = 3×A Add A to both sides of the above equation: B - A + A = 3×A + A which makes B = 4×A
Hint #4
In eq.6, replace D with 2×A, and B with 4×A: F - 2×A = 4×A + 2×A which becomes F - 2×A = 6×A Add 2×A to both sides of the equation above: F - 2×A + 2×A = 6×A + 2×A which makes F = 8×A
Hint #5
In eq.5, substitute 8×A for F: E = A + 8×A which makes E = 9×A
Solution
Substitute 4×A for B, 3×A for C, 2×A for D, 9×A for E, and 8×A for F in eq.1: A + 4×A + 3×A + 2×A + 9×A + 8×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 4×A = 4 × 1 = 4 C = 3×A = 3 × 1 = 3 D = 2×A = 2 × 1 = 2 E = 9×A = 9 × 1 = 9 F = 8×A = 8 × 1 = 8 and ABCDEF = 143298