Puzzle for September 5, 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
In eq.4, replace E with B + D (from eq.2), and C with A + F (from eq.3): B + D + F = A + F + D Subtract D and F from both sides of the above equation: B + D + F - D - F = A + F + D - D - F which simplifies to B = A
Hint #2
In eq.6, replace B with A: D - A = A + A which becomes D - A = 2×A Add A to both sides of the equation above: D - A + A = 2×A + A which makes D = 3×A
Hint #3
In eq.2, substitute A for B, and 3×A for D: E = A + 3×A which makes E = 4×A
Hint #4
In eq.5, substitute A for B, and 3×A for D: F - A = A + 3×A which becomes F - A = 4×A Add A to both sides of the above equation: F - A + A = 4×A + A which makes F = 5×A
Hint #5
Substitute 5×A for F in eq.3: C = A + 5×A which makes C = 6×A
Solution
Substitute A for B, 6×A for C, 3×A for D, 4×A for E, and 5×A for F in eq.1: A + A + 6×A + 3×A + 4×A + 5×A = 20 which simplifies to 20×A = 20 Divide both sides of the above equation by 20: 20×A ÷ 20 = 20 ÷ 20 which means A = 1 making B = A = 1 C = 6×A = 6 × 1 = 6 D = 3×A = 3 × 1 = 3 E = 4×A = 4 × 1 = 4 F = 5×A = 5 × 1 = 5 and ABCDEF = 116345